problem solving for inverse matrix

Inverse of a Matrix

Please read our Introduction to Matrices first.

What is the Inverse of a Matrix?

Just like a number has a reciprocal ...

And there are other similarities:

When we multiply a number by its reciprocal we get 1 :

When we multiply a matrix by its inverse we get the Identity Matrix (which is like "1" for matrices):

Same thing when the inverse comes first:

Identity Matrix

We just mentioned the "Identity Matrix". It is the matrix equivalent of the number "1":

A 3x3 Identity Matrix

• It is "square" (has same number of rows as columns),
• It has 1 s on the diagonal and 0 s everywhere else.
• Its symbol is the capital letter I .

The Identity Matrix can be 2×2 in size, or 3×3, 4×4, etc ...

Here is the definition:

The inverse of A is A -1 only when:

AA -1 = A -1 A = I

Sometimes there is no inverse at all.

(Note: writing AA -1 means A times A -1 )

OK, how do we calculate the inverse?

Well, for a 2x2 matrix the inverse is:

In other words: swap the positions of a and d, put negatives in front of b and c, and divide everything by ad−bc .

Note: ad−bc is called the determinant .

Let us try an example:

How do we know this is the right answer?

Remember it must be true that: AA -1 = I

So, let us check to see what happens when we multiply the matrix by its inverse:

And, hey!, we end up with the Identity Matrix! So it must be right.

It should also be true that: A -1 A = I

Why don't you have a go at multiplying these? See if you also get the Identity Matrix:

Why Do We Need an Inverse?

Because with matrices we don't divide ! Seriously, there is no concept of dividing by a matrix.

But we can multiply by an inverse , which achieves the same thing.

Imagine we can't divide by numbers ...

... and someone asks "How do I share 10 apples with 2 people?"

But we can take the reciprocal of 2 (which is 0.5), so we answer:

10 × 0.5 = 5

They get 5 apples each.

The same thing can be done with matrices:

Say we want to find matrix X, and we know matrix A and B:

It would be nice to divide both sides by A (to get X=B/A), but remember we can't divide .

But what if we multiply both sides by A -1 ?

XAA -1 = BA -1

And we know that AA -1 = I, so:

We can remove I (for the same reason we can remove "1" from 1x = ab for numbers):

And we have our answer (assuming we can calculate A -1 )

In that example we were very careful to get the multiplications correct, because with matrices the order of multiplication matters. AB is almost never equal to BA.

A Real Life Example: Bus and Train

A group took a trip on a bus , at $3 per child and $3.20 per adult for a total of $118.40.

They took the train back at $3.50 per child and $3.60 per adult for a total of $135.20.

How many children, and how many adults?

First, let us set up the matrices (be careful to get the rows and columns correct!):

This is just like the example above:

So to solve it we need the inverse of "A":

Now we have the inverse we can solve using:

There were 16 children and 22 adults!

The answer almost appears like magic. But it is based on good mathematics.

Calculations like that (but using much larger matrices) help Engineers design buildings, are used in video games and computer animations to make things look 3-dimensional, and many other places.

It is also a way to solve Systems of Linear Equations .

The calculations are done by computer, but the people must understand the formulas.

Order is Important

Say that we are trying to find "X" in this case:

This is different to the example above! X is now after A.

With matrices the order of multiplication usually changes the answer. Do not assume that AB = BA, it is almost never true.

So how do we solve this one? Using the same method, but put A -1 in front:

A -1 AX = A -1 B

And we know that A -1 A= I, so:

IX = A -1 B

We can remove I:

Why don't we try our bus and train example, but with the data set up that way around.

It can be done that way, but we must be careful how we set it up.

This is what it looks like as AX = B :

It looks so neat! I think I prefer it like this.

Also note how the rows and columns are swapped over ("Transposed") compared to the previous example.

To solve it we need the inverse of "A":

It is like the inverse we got before, but Transposed (rows and columns swapped over).

Now we can solve using:

Same answer: 16 children and 22 adults.

So matrices are powerful things, but they do need to be set up correctly!

The Inverse May Not Exist

First of all, to have an inverse the matrix must be "square" (same number of rows and columns).

But also the determinant cannot be zero (or we end up dividing by zero). How about this:

24−24? That equals 0, and 1/0 is undefined . We cannot go any further! This matrix has no Inverse.

Such a matrix is called "Singular", which only happens when the determinant is zero.

And it makes sense ... look at the numbers: the second row is just double the first row, and does not add any new information .

And the determinant 24−24 lets us know this fact.

(Imagine in our bus and train example that the prices on the train were all exactly 50% higher than the bus: so now we can't figure out any differences between adults and children. There needs to be something to set them apart.)

Bigger Matrices

The inverse of a 2x2 is easy ... compared to larger matrices (such as a 3x3, 4x4, etc).

For those larger matrices there are three main methods to work out the inverse:

• Inverse of a Matrix using Elementary Row Operations (Gauss-Jordan)
• Inverse of a Matrix using Minors, Cofactors and Adjugate
• Use a computer (such as the Matrix Calculator )
• The inverse of A is A -1 only when AA -1 = A -1 A = I
• To find the inverse of a 2x2 matrix: swap the positions of a and d, put negatives in front of b and c, and divide everything by the determinant (ad-bc).
• Sometimes there is no inverse at all

• Math Article

Inverse Matrix

• Inverse matrix 2×2 Example
• Inverse matrix 3×3 Example
• Practice problems

Matrix Inverse

If A is a non-singular square matrix, there is an existence of n x n matrix A -1 , which is called the inverse matrix of A such that it satisfies the property:

AA -1 = A -1 A = I, where I is  the Identity matrix

The identity matrix for the 2 x 2 matrix is given by

Learn: Identity matrix

It is noted that in order to find the inverse matrix, the square matrix should be non-singular whose determinant value does not equals to zero.

Let us take the square matrix A

Where a, b, c, and d represents the number.

The determinant of the matrix A is written as ad-bc, where the  value of determinant should not equal to zero for the existence of inverse . The inverse matrix can be found for 2× 2, 3× 3, …n × n matrices. Finding the inverse of a 3×3 matrix is a bit more difficult than finding the inverses of a 2 ×2 matrix .

Inverse Matrix Method

The inverse of a matrix  can be found using the three different methods. However, any of these three methods will produce the same result.

Similarly, we can find the inverse of a 3×3 matrix by finding the determinant value of the given matrix.

Check out : Inverse matrix calculator

One of the most important methods of finding the matrix inverse involves finding the minors and cofactors of elements of the given matrix. Observe the below steps to understand this method clearly.

• The inverse matrix is also found using the following equation:

A -1 = adj(A)/det(A),

          w here  adj(A) refers to the adjoint of a matrix A,  det(A) refers to the determinant of a matrix A.

• The adjoint of a matrix A or adj(A) can be found using the following method.

          In order to find the adjoint of a matrix A first, find the cofactor matrix of a given matrix and then   

          take the transpose of a cofactor matrix.

• The cofactor of a matrix can be obtained as

C ij  = (-1) i+j  det (M ij )

Here, M ij refers to the (i,j) th   minor matrix after removing the i th row and the j th column. You can also say that the transpose of a cofactor matrix is also called the adjoint of a matrix A.

Learn how to find the adjoint of a matrix here.

Similarly, we can also find the inverse of a 3 x 3 matrix . Here also the first step would be to find the determinant, followed by the next step – Transpose.

Finding an Inverse Matrix by Elementary Transformation

Let us consider three matrices X, A and B such that X = AB. To determine the inverse of a matrix using elementary transformation, we convert the given matrix into an identity matrix.  Learn more about  how to do elementary transformations of matrices here.

If the inverse of matrix A, A -1 exists then to determine A -1 using elementary row operations

• Write A = IA, where I is the identity matrix of the same order as A.
• Apply a sequence of row operations till we get an identity matrix on the LHS and use the same elementary operations on the RHS to get I = BA. The matrix B on the RHS is the inverse of matrix A.
• To find the inverse of A using column operations, write A = IA and apply column operations sequentially till I = AB is obtained, where B is the inverse matrix of A.

Click here to understand the method of finding the inverse of a matrix using elementary operations .

Inverse of a Matrix Formula

Let  \(\begin{array}{l}A=\begin{bmatrix} a &b \\ c & d \end{bmatrix}\end{array} \) be  the 2 x 2 matrix. The inverse matrix of A is given by the formula,

Let  \(\begin{array}{l}A=\begin{bmatrix} a_{11} &a_{12} & a_{13}\\ a_{21} &a_{22} &a_{23} \\ a_{31} & a_{32} & a_{33} \end{bmatrix}\end{array} \) be  the 3 x 3 matrix. The inverse matrix is:

Inverse Matrix 2 x 2 Example

To understand this concept better let us take a look at the following example.

Example : Find the inverse of matrix A given below:

Inverse Matrix 3 x 3 Example

Determinant of the given matrix is

Let us find the minors of the given matrix as given below:

Now, find the adjoint of a matrix by taking the transpose of cofactors of the given matrix.

A -1 = (1/|A|) Adj A

Hence, the inverse of the given matrix is:

A few important properties of the inverse matrix are listed below.

• If A is nonsingular, then (A -1 ) -1   =  A
• If A and B are nonsingular matrices, then AB is nonsingular. Thus, (AB) -1   =  B -1 A -1
• If A is nonsingular then (A T ) -1   =  (A -1 ) T
• If A is any matrix and A -1  is its inverse, then AA -1  = A -1 A = I n , where n is the order of matrices

Practice Problems

• Find the inverse of a matrix: \(\begin{array}{l}A = \begin{bmatrix} 1 & 2 &3 \\ 3 & -2 &1 \\ 4 & 1 & 1 \end{bmatrix}\end{array} \)
• Obtain the inverse of the matrix using elementary operations: \(\begin{array}{l} A = \begin{bmatrix} 0 & 1 &2 \\ 1 & 2 &3 \\ 3 & 1 & 1 \end{bmatrix}\end{array} \)  
• Find the inverse of a matrix: \(\begin{array}{l}\begin{bmatrix} 1 & 2\\ 3 & 4 \end{bmatrix}\end{array} \)

To learn more about matrix and inverse of a matrix download BYJU’S- The Learning App.

Frequently Asked Questions – FAQs

What is concept inverse of a matrix, how do you find the inverse of a 3×3 matrix, is adjoint and inverse the same, how to do you know whether the given matrix has inverse, what are the properties of inverse matrix.

Leave a Comment Cancel reply

Your Mobile number and Email id will not be published. Required fields are marked *

Request OTP on Voice Call

Post My Comment

Good Teacher

Register with BYJU'S & Download Free PDFs

Register with byju's & watch live videos.

Free Mathematics Tutorials

• Math Problems
• Algebra Questions and Problems
• Graphs of Functions, Equations, and Algebra
• Free Math Worksheets to Download
• Analytical Tutorials
• Solving Equation and Inequalities
• Online Math Calculators and Solvers
• Free Graph Paper
• Math Software
• The Applications of Mathematics in Physics and Engineering
• Exercises de Mathematiques Utilisant les Applets
• Calculus Tutorials and Problems
• Calculus Questions With Answers
• Free Calculus Worksheets to Download
• Geometry Tutorials and Problems
• Online Geometry Calculators and Solvers
• Free Geometry Worksheets to Download
• Trigonometry Tutorials and Problems for Self Tests
• Free Trigonometry Questions with Answers
• Free Trigonometry Worksheets to Download
• Elementary Statistics and Probability Tutorials and Problems
• Mathematics pages in French
• About the author
• Primary Math
• Middle School Math
• High School Math
• Free Practice for SAT, ACT and Compass Math tests

Inverse Matrix Questions with Solutions

Tutorials including examples and questions with detailed solutions on how to find the inverse of square matrices using the method of the row echelon form and the method of cofactors. The properties of inverse matrices are discussed and various questions, including some challenging ones, related to inverse matrices are included along with their detailed solutions.

Page Content

Definition of the identity matrix, definition of the inverse of a matrix, find the inverse of a square matrix using the row reduction method, find the inverse of a square matrix using minors, cofactors and adjugate, formula for the inverse of a 2 by 2 matrix, properties of inverse matrices, questions on inverse matrices.

• Solutions to the Questions

Example 2 Find the inverse of matrix A given by \[ A = \begin{bmatrix} 1&1 \\ 2&4 \end{bmatrix} \] if it exists. Solution Write the augmented matrix \( [ A | I )\) \[ \begin{bmatrix} 1&1&|&1&0\\2&4&|&0&1 \end{bmatrix} \] step 1 \[ \color{red}{\begin{matrix} \\ R_2 - 2 \times R_1 \end{matrix} } \begin{bmatrix} 1&1&|&1&0\\0&2&|&-2&1 \end{bmatrix} \] step 2 \[ \color{red}{\begin{matrix} \\ (1/2)R_2 \end{matrix} } \begin{bmatrix} 1&1&|&1&0\\0&1&|&-1&1/2 \end{bmatrix} \] step 3 \[ \color{red}{\begin{matrix} R_1 - R_2 \\ \\ \end{matrix} } \begin{bmatrix} 1&0&|&2&-1/2\\0&1&|&-1&1/2 \end{bmatrix} \] The inverse of A is the 2 × 2 matrix on the right side given by \[ A^{-1} = \begin{bmatrix} 2&-1/2\\-1&1/2 \end{bmatrix} \]

Example 3 Find the inverse of matrix A given by \[ A = \begin{bmatrix}-2&2&0 \\ 2&1&3\\ -2&4&-2\end{bmatrix} \] if it exists. Solution Write the augmented matrix \( [ A | I )\) \[ \begin{bmatrix} -2&2&0&|&1&0&0\\ 2&1&3&|&0&1&0 \\ -2 & 4 & -2 &|& 0 & 0 & 1 \end{bmatrix} \] step 1 \[ \color{red}{ \begin{matrix} \\ R_2 + R_1 \\ R_3 - R_1 \end{matrix} } \begin{bmatrix} -2&2&0&|&1&0&0\\ 0&3&3&|&1&1&0 \\ 0 & 2 & -2 &|& -1 & 0 & 1 \end{bmatrix} \] step 2 \[ \color{red}{ \begin{matrix} \\ \\ R_3 - (2/3) R_2 \\ \end{matrix} } \begin{bmatrix} -2&2&0&|&1&0&0\\ 0&3&3&|&1&1&0 \\ 0 & 0 & - 4&|& -5/3 & -2/3 & 1 \end{bmatrix} \] step 3 \[ \color{red}{ \begin{matrix} \\ \\ (-1/4)R_3 \\ \end{matrix}} \begin{bmatrix} -2&2&0&|&1&0&0\\ 0&3&3&|&1&1&0 \\ 0 & 0 & 1&|& 5/12 & 1/6 & -1/4 \end{bmatrix} \] step 4 \[ \color{red}{ \begin{matrix} \\ R_2 - 3\times R_3 \\ \\ \end{matrix} } \begin{bmatrix} -2&2&0&|&1&0&0\\ 0&3&0&|&-1/4&1/2&3/4 \\ 0 & 0 & 1&|& 5/12 & 1/6 & -1/4 \end{bmatrix} \] step 5 \[ \color{red}{ \begin{matrix} \\ (1/3) R_2 \\ \\ \end{matrix}} \begin{bmatrix} -2&2&0&|&1&0&0\\ 0&1&0&|&-1/12&1/6&1/4 \\ 0 & 0 & 1&|& 5/12 & 1/6 & -1/4 \end{bmatrix} \] step 6 \[ \color{red}{ \begin{matrix} R_1- 2\times R_2 \\ \\ \\ \end{matrix} } \begin{bmatrix} -2&0&0&|&7/6&-1/3&-1/2\\ 0&1&0&|&-1/12&1/6&1/4 \\ 0 & 0 & 1&|& 5/12 & 1/6 & -1/4 \end{bmatrix} \] step 7 \[ \color{red}{ \begin{matrix} (-1/2) R_1 \\ \\ \\ \end{matrix} } \begin{bmatrix} 1&0&0&|&-7/12&1/6&1/4\\ 0&1&0&|&-1/12&1/6&1/4 \\ 0 & 0 & 1&|& 5/12 & 1/6 & -1/4 \end{bmatrix} \] Hence \[ A^{-1} = \begin{bmatrix} -7/12&1/6&1/4\\ -1/12&1/6&1/4 \\ 5/12 & 1/6 & -1/4 \end{bmatrix} \] More examples on how to find matrix inverse using row operations are included.

This method is explained using a numerical example. Matrix A is given below. \[ A = \begin{bmatrix} -1&0&1\\ 2&-1&2 \\ -1 & 2 & 1 \end{bmatrix} \] a) Find the matrices of minors and cofactors, the adjugate and the inverse of A. Matrix of Minors The entry \( M_{i,j} \) of the matrix of minors of matrix A is given by the determinant obtained by deleting the \( i^{th}\) row and the \( j^{th}\) column. To find \( M_{1,1} \), delete row 1 and column 1 from matrix A and find the determinant of the remaining 2 by 2 matrix as follows: \( M_{1,1} = Det \begin{bmatrix} .&.&.\\ .&-1&2 \\ .& 2 & 1 \end{bmatrix} = -1 - 4 = -5\) To find \( M_{1,2} \), delete row 1 and column 2 from matrix A and find the determinant of the remaining 2 by 2 matrix as follows: \( M_{1,2} = Det \begin{bmatrix} .&.&.\\ 2&.&2 \\ -1 & . & 1 \end{bmatrix} = 2 -(-2) = 4 \) To find \( M_{1,3} \), delete row 1 and column 3 from matrix A and find the determinant of the remaining 2 by 2 matrix as follows: \( M_{1,3} = Det \begin{bmatrix} .&.&.\\ 2&-1&. \\ -1 & 2 & . \end{bmatrix} = 4 - 1 = 3 \) To find \( M_{2,1} \), delete row 2 and column 1 from matrix A and find the determinant of the remaining 2 by 2 matrix as follows: \( M_{2,1} = Det \begin{bmatrix} .&0&1\\ .&.&. \\ . & 2 & 1 \end{bmatrix} = 0 - 2 = - 2 \) ... ... The remaining entries are given by: \( M_{2,2} = 0 \) , \( M_{2,3} = -2 \) , \( M_{3,1} = 1\) , \( M_{3,2} = -4\) , \( M_{3,3} = 1\). The matrix of minors M is given by \( M = \begin{bmatrix} -5&4&3\\ -2&0&-2\\ 1&-4&1 \end{bmatrix} \) Matrix of Cofactors The entries \( C_{i,j} \) of the matrix of cofactors C of matrix A are given by \( C_{i,j} = (-1)^{i+j}M{i,j} \) An evaluation of the entries \( C_{i,j} \) gives: \( C_{1,1} = (-1)^{1+1} M_{1,1} = -5 \) \( C_{1,2} = (-1)^{1+2} M_{1,2} = - 4 \) \( C_{1,3} = (-1)^{1+3} M_{1,3} = 3 \) \( C_{2,1} = (-1)^{2+1} M_{2,1} = 2 \) \( C_{2,2} = (-1)^{2+2} M_{2,2} = 0 \) \( C_{3,1} = (-1)^{3+1} M_{3,1} = 1 \) \( C_{3,2} = (-1)^{3+2} M_{3,2} = 4 \) \( C_{3,3} = (-1)^{3+3} M_{3,1} = 1 \) Hence the matrix C of cofactors is given by \( C = \begin{bmatrix} -5&-4&3\\ 2&0&2\\ 1&4&1 \end{bmatrix} \) Adjugate (or adjunct) of a Matrix The adjugate (or adjunct) of matrix A is the transpose of its matrix of cofactors C. \( Adjugate(A) = C^T = \begin{bmatrix} -5&2&1\\ -4&0&4\\ 3&2&1 \end{bmatrix} \) Inverse Matrix We now need to find the determinant D of matrix A. Using the first row of matrix A and the corresponding minors already found, D is given by \( D = det\begin{bmatrix} -1&0&1\\ 2&-1&2 \\ -1 & 2 & 1 \end{bmatrix} = A_{11}M_{1,1} - A_{1,2}M_{1,2} + A_{1,3}M_{1,3} = 8\) The inverse of \( A \) is given by \( A^{-1} = \dfrac{1}{D} C^T = \dfrac{1}{8} \begin{bmatrix} -5&2&1\\ -4&0&4\\ 3&2&1 \end{bmatrix} = \begin{bmatrix} -\dfrac{5}{8}&\dfrac{1}{4}&\dfrac{1}{8}\\ -\dfrac{1}{2}&0&\dfrac{1}{2}\\ \dfrac{3}{8}&\dfrac{1}{4}&\dfrac{1}{8}\end{bmatrix}\)

Using any of the two methods described above, it can be shown that the inverse of matrix A given by \[ A = \begin{bmatrix} a & b\\ c & d \\ \end{bmatrix} \] is given by \[ A^{-1} = \dfrac{1}{ad - bc}\begin{bmatrix} d & -b\\ -c & a \\ \end{bmatrix} \]

A matrix that has an inverse is called an invertible matrix.

• If A is an invertible matrix, its inverse is unique.
• \( A A^{-1} = A^{-1} A = I \)
• If matrices A and B are invertible, then:\( (AB)^{-1} = B^{-1}A^{-1} \)
• A matrix is invertible if and only if its determinant is not equal to zero.
• A matrix whose determinant is not equal to zero is called nonsingular.
• \( (A^T)^{-1} = (A^{-1})^T \)
• \( Det(A^{-1}) = \dfrac{1}{Det(A)} \)
• \( (A^{-1})^{-1} = A \)
• Question 1 Use row reduction method to find the inverse of the following matrices: \( A = \begin{bmatrix} -1&-1&1\\ 2&0&-2 \\ 1 & 1 & 1 \end{bmatrix} \) , \( B = \begin{bmatrix} 1&0&1&2\\ -1& 1 & 2 & 0 \\ -2& 0 & 1 & 2 \\ 0 & 0 & 0 & 1 \end{bmatrix} \)
• Question 2 Use the method of cofactors to find the inverse of the following matrix. \( A = \begin{bmatrix} -1&0&3\\ 3&2&2 \\ 0& 0 & 1 \end{bmatrix} \)
• Question 3 A, B and C are 2 by 2 matrices. Matrices B and C are given by: \[ B = \begin{bmatrix} -1&-1\\ -2& 1 \end{bmatrix} , C = \begin{bmatrix} 2 & -1\\ -2 & 2 \end{bmatrix} \] Find matrix A such that AB = C.
• Question 4 For what value(s) of k is each of the matrices given below invertible? a) \( \begin{bmatrix} k & -1 & 4\\ 2 & 0 & 1\\ -1 & 0 & -1 \end{bmatrix} \) , b) \( \begin{bmatrix} k & -1 \\ -1 & 3 \end{bmatrix} \) , c) \( \begin{bmatrix} k & -1 & 4\\ 0 & k + 1 & 1\\ 0 & 0 & k -3 \end{bmatrix} \)
• Question 6 Matrix A is given by \( A = \begin{bmatrix} a & 0 & 0 & 0\\ 0 & b & 0 & 0 \\ 0 & 0 & c & 0\\ 0 & 0 & 0 & d \end{bmatrix} \) Find a formula for the inverse of matrix A if none of the parameters a, b, c and d is equal to zero.
• Question 7 Use the inverse matrix to solve the system of equations \( \begin{bmatrix} 1&0&1&2\\ -1& 1 & 2 & 0 \\ -2& 0 & 1 & 2 \\ 0 & 0 & 0 & 1 \end{bmatrix}\begin{bmatrix} x_1\\ x_2 \\ x_3\\ x_4 \end{bmatrix} = \begin{bmatrix} 0\\ 1 \\ -1\\ 2 \end{bmatrix} \)
• Question 8 What is the most efficient method to solve the following systems of equations? \( A X_1 = B_1 \) , \( A X_2 = B_2 \) , \( A X_3 = B_3 \) ... \( A X_i = B_i \)
• Question 9 A and B are invertible matrices of the same dimension related by: \( A^{-1} = A B \). Find B in terms of A or its inverse.
• Question 10 1) Give an example of 2 by 2 matrices A and B such that neither A nor B are invertible yet A + B is invertible. 2) Give an example of 2 by 2 matrices A and B such that neither A nor B are invertible yet A - B is invertible.
• Question 11 Use any of the two methods to find a formula for the inverse of a 2 by 2 matrix.(It is already given above without proof).
• Question 12 The system of equations in matrix form \( A X = B \) has the following solutions: \( X_1 = \begin{bmatrix} -1\\ 2 \\ 3 \end{bmatrix} \) for \( B_1 = \begin{bmatrix} 2\\ 13 \\ 3 \end{bmatrix} \) , \( X_2 = \begin{bmatrix} 0\\ -1 \\ 1 \end{bmatrix} \) for \( B_2 = \begin{bmatrix} 4\\ 2 \\ 2 \end{bmatrix} \) , \( X_3 = \begin{bmatrix} 1\\ 1 \\ 1 \end{bmatrix} \) for \( B_3 = \begin{bmatrix} 4\\ 5 \\ 3 \end{bmatrix} \). Find X for \( B = \begin{bmatrix} 1\\ -9 \\ -1 \end{bmatrix} \).

Solutions to the Above Questions

• Solution to Question 2 We first find the minors. \( M_{1,1} = Det \begin{bmatrix} .&.&.\\ .&2&2 \\ .& 0 & 1 \end{bmatrix} = 2\) , \( M_{1,2} = Det \begin{bmatrix} .&.&.\\ 3&.&2 \\ 0& . & 1 \end{bmatrix} = 3\) , \( M_{1,3} = Det \begin{bmatrix} .&.&.\\ 3&2&. \\ 0& 0 & . \end{bmatrix} = 0 \) \( M_{2,1} = Det \begin{bmatrix} .&0&3\\ .&.&. \\ .& 0 & 1 \end{bmatrix} = 0\) , \( M_{2,2} = Det \begin{bmatrix} -1&.&3\\ . & . & . \\ 0& . & 1 \end{bmatrix} = -1\) , \( M_{2,3} = Det \begin{bmatrix} -1&0&.\\ . & . & .\\ 0& 0 & . \end{bmatrix} = 0\) \( M_{3,1} = Det \begin{bmatrix} .&0&3\\ .&2&2 \\ . & . & . \end{bmatrix} = - 6\) , \( M_{3,2} = Det \begin{bmatrix} -1&.&3\\ 3&.&2 \\ . & . & . \end{bmatrix} = - 11\) , \( M_{3,3} = Det \begin{bmatrix} -1&0& .\\ 3&2& . \\ . & . & . \end{bmatrix} = - 2\) Matrix C of cofactors whose entries defined are defined as \( C_{i,j} = (-1)^{i+j} M_{i,j} \) \[ C = \begin{bmatrix} 2&-3&0\\ 0&-1&0 \\ - 6 & 11 & -2 \end{bmatrix} \] We need to find D the determinant of A using the third row (it has 2 zeros!) \( D = A_{3,3} M_{3,3} = - 2 \) The inverse of A is given by \( A^{-1} = \dfrac{1}{D} C^T = -\dfrac{1}{2} \begin{bmatrix} 2&0&-6\\ -3&-1&11\\ 0&0&-2 \end{bmatrix} = \begin{bmatrix} -1&0&3\\ \dfrac{3}{2}& \dfrac{1}{2} & -\dfrac{11}{2}\\ 0&0&1\end{bmatrix}\)
• Solution to Question 3 Given \( A B = C \) Right multiply both sides by \( B^{-1} \) \( A B B^{-1} = C B^{-1}\) Use associativity on the left side \( A (B B^{-1}) = C B^{-1} \) Simplify \( A I = C B^{-1} \) \( A = C B^{-1} \) Use the formula for the inverse of a 2 by 2 matrix to find the inverse of B. \( Det(B) = -3 \) \( B^{-1} = - \dfrac{1}{3} \begin{bmatrix} 1&1\\ 2& -1 \end{bmatrix} \) \( A = C B^{-1} = \begin{bmatrix} 2 & -1\\ -2 & 2 \end{bmatrix} (- \dfrac{1}{3}) \begin{bmatrix} 1&1\\ 2& -1 \end{bmatrix} = \begin{bmatrix} 0&-1\\ -2/3& 4/3 \end{bmatrix}\) Note: you may check the answer for matrix A by substituting in the equation \( A B = C \)
• Solution to Question 4 A matrix is invertible if its determinant is not equal to zero. a) Using the second column, Det\( \begin{bmatrix} k & -1 & 4\\ 2 & 0 & 1\\ -1 & 0 & -1 \end{bmatrix} = - 1\) The matrix is invertible for any k real b) Det\( \begin{bmatrix} k & -1 \\ -1 & 3 \end{bmatrix} = 3k - 1\) \( 3k - 1 \ne 0 \) \( k \ne 1/3 \) The matrix in part b) is invertible for all real values of k not equal to 1/3. c) The given matrix is an upper triangular matrix and its determinant is equal to the product of the terms in the diagonal left to right. Det \( \begin{bmatrix} k & -1 & 4\\ 0 & k + 1 & 1\\ 0 & 0 & k -3 \end{bmatrix} k(k+1)(k-3)\) \( k(k+1)(k-3) \ne 0 \) The given matrix is invertible if k is not equal to 0, - 1 or 3.
• Solution to Question 5 Right multiply the two sides of the equation by \( S^{-1} \) \( P S^{-1} = Q R^{-1} S S^{-1} \) simplify \( P S^{-1} = Q R^{-1} I \) \( P S^{-1} = Q R^{-1} \) Left multiply the two sides of the equation by \( Q^{-1} \) \( Q^{-1} P S^{-1} = Q^{-1} Q R^{-1} \) simplify \( Q^{-1} P S^{-1} = I R^{-1}\) \( Q^{-1} P S^{-1} = R^{-1} \) Take the inverse of both sides \( (Q^{-1} P S^{-1})^{-1} = (R^{-1})^{-1} \) Simplify \( R = S P^{-1} Q \)
• Solution to Question 6 Write the augmented matrix \( [ A | I ]\) \( \begin{bmatrix} a & 0 & 0 & 0&|&1&0&0&0\\ 0 & b & 0 & 0&|&0&1&0&0 \\ 0 & 0 & c & 0 &|& 0 & 0 & 1 & 0\\ 0 & 0 & 0 & d &|& 0 & 0 & 0 & 1 \end{bmatrix} \) Multiply row (1) by 1/a, row (2) by 1/b, row (3) by 1/c and row (4) by 1/d and simplify \( \begin{bmatrix} 1 & 0 & 0 & 0&|&1/a&0&0&0\\ 0 & 1 & 0 & 0&|&0&1/b&0&0 \\ 0 & 0 & 1 & 0 &|& 0 & 0 & 1/c & 0\\ 0 & 0 & 0 & 1 &|& 0 & 0 & 0 & 1/d \end{bmatrix} \) The inverse of the given matrix is \( A^{-1} = \begin{bmatrix} 1/a&0&0&0\\ 0&1/b&0&0 \\ 0 & 0 & 1/c & 0\\ 0 & 0 & 0 & 1/d \end{bmatrix} \)
• Solution to Question 7 The system is of the form A X = B with A = \( \begin{bmatrix} 1&0&1&2\\ -1& 1 & 2 & 0 \\ -2& 0 & 1 & 2 \\ 0 & 0 & 0 & 1 \end{bmatrix} \) , \( B =\begin{bmatrix} 0\\ 1 \\ -1\\ 2 \end{bmatrix} \) and \( X = \begin{bmatrix} x_1\\ x_2 \\ x_3\\ x_4 \end{bmatrix} \) Right multiply both sides of the equation by \( A^{-1} \) and simplify. \( A^{-1} A X = A^{-1} B \) \( I_3 X = A^{-1} B , I_3 \) is the 3 by 3 identity matrix Simplify the above \( X = A^{-1} B \) The inverse of matrix A was calculated in question 1 and is given by (it is matrix B in question 1) \( A^{-1} = \begin{bmatrix} 1/3&0&-1/3&0\\ -1&1&-1&4 \\ 2/3 & 0 & 1/3 & -2\\ 0 & 0 & 0 & 1 \end{bmatrix} \) \( \begin{bmatrix} x_1\\ x_2 \\ x_3\\ x_4 \end{bmatrix} = A^{-1} B = \begin{bmatrix} 1/3&0&-1/3&0\\ -1&1&-1&4 \\ 2/3 & 0 & 1/3 & -2\\ 0 & 0 & 0 & 1 \end{bmatrix} \begin{bmatrix} 0\\ 1 \\ -1\\ 2 \end{bmatrix} = \begin{bmatrix}\dfrac{1}{3}\\ 10\\ -\dfrac{13}{3}\\ 2\end{bmatrix}\)
• Solution to Question 8 Since matrix A is common to all the given systems, the most efficient method solving systems of equations of the form \( A X_1 = B_1 \) , \( A X_2 = B_2 \) , \( A X_3 = B_3 \) ... \( A X_2 = B_i \) is to find the inverse of matrix A and solve as follows (see question 7 above) \( X_1 = A^{-1} B_1 \) , \( X_2 = A^{-1} B_2 \) , \( X_3 = A^{-1} B_3 \) ... \( X_i = A^{-1} B_i \)
• Solution to Question 10 There are many possible answers to both parts of this question. 1) \(A = \begin{bmatrix} 1 & 1\\ 0 & 0 \\ \end{bmatrix} , B = \begin{bmatrix} 0 & 0\\ - 1 & 1 \\ \end{bmatrix} \) , \(A + B = \begin{bmatrix} 1 & 1\\ - 1 & 0 \\ \end{bmatrix} \) 2) \(A = \begin{bmatrix} 3 & 1\\ 0 & 0 \\ \end{bmatrix} , B = \begin{bmatrix} 0 & 0\\ - 1 & 4 \\ \end{bmatrix} \) , \(A - B = \begin{bmatrix} 3 & 1\\ 1 & - 4 \\ \end{bmatrix} \) Check that the determinant of matrices A and B are equal to zero and therefore non invertible. Check that the determinants of A + B and A - B are not equal to zero and therefore invertible.
• Solution to Question 11 Let \(A = \begin{bmatrix} a & b\\ c & d \\ \end{bmatrix} \) We shall use the method of cofactors. We first calculate the minors \( M_{1,1} = d\) , \( M_{1,2} = c\) , \( M_{2,1} = b\) , \( M_{2,2} = a\) Then the cofactors using the formula: \( C_{i,j} = (-1)^{i+j}M_{i,j} \) \( C_{1,1} = d\) , \( C_{1,2} = - c\) , \( C_{2,1} = - b\) , \( C_{2,2} = a\) The determinant of A is \( D = a d - b c \) \( A^{-1} = \dfrac{1}{a d - b c} \begin{bmatrix} d & - c\\ - d & a \\ \end{bmatrix}^T = \dfrac{1}{a d - b c} \begin{bmatrix} d & - d\\ - c & a \\ \end{bmatrix} \)
• Solution to Question 12 The solutions may be written in matrix form as follows \( A \begin{bmatrix} -1 & 0 & 1 \\ 2 & - 1 & 1\\ 3 & 1 & 1 \end{bmatrix} = \begin{bmatrix} 5 & 4 & 4\\ 17 & 2 & 5\\ 5 & 2 & 3 \end{bmatrix} \) which gives \( A = \begin{bmatrix} 5 & 4 & 4\\ 17 & 2 & 5\\ 5 & 2 & 3 \end{bmatrix} \begin{bmatrix} -1 & 0 & 1 \\ 2 & - 1 & 1\\ 3 & 1 & 1 \end{bmatrix}^{-1} \) which gives \( A^{-1} = \begin{bmatrix} -1 & 0 & 1 \\ 2 & - 1 & 1\\ 3 & 1 & 1 \end{bmatrix} \begin{bmatrix} 5 & 4 & 4\\ 17 & 2 & 5\\ 5 & 2 & 3 \end{bmatrix}^{-1} \) The solution X is given by \( X = A^{-1} B = \begin{bmatrix} -1 & 0 & 1 \\ 2 & - 1 & 1\\ 3 & 1 & 1 \end{bmatrix} \begin{bmatrix} 5 & 4 & 4\\ 17 & 2 & 5\\ 5 & 2 & 3 \end{bmatrix}^{-1} \begin{bmatrix} 1\\ -9 \\ -1 \end{bmatrix} = \begin{bmatrix} 1\\ -2 \\ -1 \end{bmatrix} \)

More References and Links to Matrices

• Matrices with Examples and Questions with Solutions .
• Find the Inverse of a Matrix Using Row Reduction
• multiplication of matrices using an applet .
• Find Inverse Matrix - Calculator.
• Find Inverse of 3 by 3 Matrix - Calculator.
• Step by Step Solver to Find the Inverse of a 3 by 3 Matrix .
• Step by Step Solver to Calculate the Determinant of a 3 by 3 Matrix .

POPULAR PAGES

privacy policy

Problems in Mathematics

• Inverse Matrices
• An $n\times n$ matrix $A$ is said to be invertible if there exists an $n\times n$ matrix $B$ such that $AB=BA=I$. Such a matrix $B$ is unique and called the inverse matrix of $A$, denoted by $A^{-1}$

Let $A, B$ be $n\times n$ matrices.

• $A$ is invertible if and only if $\rref([ A \mid I_n])=[ I_n \mid A’]$ for some $n\times n$ matrix $A’$. In this case, $A’=A^{-1}$.
• $A$ is invertible if and only if $A$ is nonsingular.
• If $A, B$ are invertible, then $(AB)^{-1}=B^{-1}A^{-1}$
• A $2\times 2$ matrix $A=\begin{bmatrix} a & b\\ c& d \end{bmatrix}$ is invertible if and only if the determinant $\det(A)=ad-bc \neq 0$. If $A$ is invertible, then the inverse matrix is given by $A^{-1}=\frac{1}{\det(A)}\begin{bmatrix} d & -b\\ -c& a \end{bmatrix}$.
• If $A$ is invertible, then $A^{\trans}$ is invertible and $(A^{\trans})^{-1}=(A^{-1})^{\trans}$.
• For each of the following $3\times 3$ matrices $A$, determine whether $A$ is invertible and find the inverse $A^{-1}$ if exists by computing the augmented matrix $[A|I]$, where $I$ is the $3\times 3$ identity matrix. (a) $A=\begin{bmatrix} 1 & 3 & -2 \\ 2 &3 &0 \\ 0 & 1 & -1 \end{bmatrix}$ (b) $A=\begin{bmatrix} 1 & 0 & 2 \\ -1 &-3 &2 \\ 3 & 6 & -2 \end{bmatrix}$.
• Let A be the matrix \[\begin{bmatrix} 1 & -1 & 0 \\ 0 &1 &-1 \\ 0 & 0 & 1 \end{bmatrix}.\] Is the matrix $A$ invertible? If not, then explain why it isn’t invertible. If so, then find the inverse. ( The Ohio State University )
• Find the inverse matrix of \[A=\begin{bmatrix} 1 & 1 & 2 \\ 0 &0 &1 \\ 1 & 0 & 1 \end{bmatrix}\] if it exists. If you think there is no inverse matrix of $A$, then give a reason. ( The Ohio State University )
• Let $A$ be the following $3\times 3$ upper triangular matrix. \[A=\begin{bmatrix} 1 & x & y \\ 0 &1 &z \\ 0 & 0 & 1 \end{bmatrix},\] where $x, y, z$ are some real numbers. Determine whether the matrix $A$ is invertible or not. If it is invertible, then find the inverse matrix $A^{-1}$.
• For which choice(s) of the constant $k$ is the following matrix invertible? \[A=\begin{bmatrix} 1 & 1 & 1 \\ 1 &2 &k \\ 1 & 4 & k^2 \end{bmatrix}.\] ( Johns Hopkins University )
• Suppose that $M, P$ are two $n \times n$ non-singular matrix. Prove that there is a matrix $N$ such that $MN = P$.
• Let $A$ be an $n \times n$ matrix satisfying $A^2+c_1A+c_0I=O$, where $c_0, c_1$ are scalars, $I$ is the $n\times n$ identity matrix, and $O$ is the $n\times n$ zero matrix. Prove that if $c_0\neq 0$, then the matrix $A$ is invertible (nonsingular). How about the converse? Namely, is it true that if $c_0=0$, then the matrix $A$ is not invertible?
• A square matrix $A$ is called idempotent if $A^2=A$. Show that a square invertible idempotent matrix is the identity matrix.
• Find the inverse matrix of $A=\begin{bmatrix} 1 & 0 & 1 \\ 1 &0 &0 \\ 2 & 1 & 1 \end{bmatrix}$ if it exists. If you think there is no inverse matrix of $A$, then give a reason. See (a)
• Find a nonsingular $2\times 2$ matrix $A$ such that $A^3=A^2B-3A^2$, where $B=\begin{bmatrix} 4 & 1\\ 2& 6 \end{bmatrix}$. Verify that the matrix $A$ you obtained is actually a nonsingular matrix. See (b)
• Determine whether there exists a nonsingular matrix $A$ if $A^2=AB+2A$, where $B$ is the following matrix. If such a nonsingular matrix $A$ exists, find the inverse matrix $A^{-1}$. (a) \[B=\begin{bmatrix} -1 & 1 & -1 \\ 0 &-1 &0 \\ 1 & 2 & -2 \end{bmatrix}\] (b) \[B=\begin{bmatrix} -1 & 1 & -1 \\ 0 &-1 &0 \\ 2 & 1 & -4 \end{bmatrix}.\]
• Determine whether there exists a nonsingular matrix $A$ if $A^4=ABA^2+2A^3$, where $B$ is the following matrix. $B=\begin{bmatrix} -1 & 1 & -1 \\ 0 &-1 &0 \\ 2 & 1 & -4 \end{bmatrix}.$ If such a nonsingular matrix $A$ exists, find the inverse matrix $A^{-1}$. ( The Ohio State University, Linear Algebra Final Exam Problem )
• Let $A, B, C$ be the following $3\times 3$ matrices. \[A=\begin{bmatrix} 1 & 2 & 3 \\ 4 &5 &6 \\ 7 & 8 & 9 \end{bmatrix}, B=\begin{bmatrix} 1 & 0 & 1 \\ 0 &3 &0 \\ 1 & 0 & 5 \end{bmatrix}, C=\begin{bmatrix} -1 & 0\ & 1 \\ 0 &5 &6 \\ 3 & 0 & 1 \end{bmatrix}.\] Then compute and simplify the following expression. \[(A^{\trans}-B)^{\trans}+C(B^{-1}C)^{-1}.\] ( The Ohio State University )
• Let $A$ be the coefficient matrix of the system of linear equations \begin{align*} -x_1-2x_2&=1\\ 2x_1+3x_2&=-1. \end{align*} (a) Solve the system by finding the inverse matrix $A^{-1}$. (b) Let $\mathbf{x}=\begin{bmatrix} x_1 \\ x_2 \end{bmatrix}$ be the solution of the system obtained in part (a). Calculate and simplify \[A^{2017}\mathbf{x}.\] ( The Ohio State University )
• (a) Let $A$ be a $6\times 6$ matrix and suppose that $A$ can be written as $A=BC$, where $B$ is a $6\times 5$ matrix and $C$ is a $5\times 6$ matrix. Prove that the matrix $A$ cannot be invertible. (b) Let $A$ be a $2\times 2$ matrix and suppose that $A$ can be written as $A=BC$, where $B$ is a $ 2\times 3$ matrix and $C$ is a $3\times 2$ matrix. Can the matrix $A$ be invertible?
• For a real number $a$, consider $2\times 2$ matrices $A, P, Q$ satisfying the following five conditions. (1) $A=aP+(a+1)Q$, (2) $P^2=P$, (3) $Q^2=Q$, (4) $PQ=O$, (5) $QP=O$, where $O$ is the $2\times 2$ zero matrix. Then do the following problems. (a) Prove that $(P+Q)A=A$. (b) Suppose $a$ is a positive real number and let $A=\begin{bmatrix} a & 0\\ 1& a+1 \end{bmatrix}$. Then find all matrices $P, Q$ satisfying conditions (1)-(5). (c) Let $n$ be an integer greater than $1$. For any integer $k$, $2\leq k \leq n$, we define the matrix $A_k=\begin{bmatrix} k & 0\\ 1& k+1 \end{bmatrix}$. Then calculate and simplify the matrix product $A_nA_{n-1}A_{n-2}\cdots A_2$. ( Tokyo University Entrance Exam 2007 )
• Let $\mathbf{v}$ be a nonzero vector in $\R^n$. Then the dot product $\mathbf{v}\cdot \mathbf{v}=\mathbf{v}^{\trans}\mathbf{v}\neq 0$. Set $a:=\frac{2}{\mathbf{v}^{\trans}\mathbf{v}}$ and define the $n\times n$ matrix $A$ by $A=I-a\mathbf{v}\mathbf{v}^{\trans}$, where $I$ is the $n\times n$ identity matrix. Prove that $A$ is a symmetric matrix and $AA=I$. Conclude that the inverse matrix is $A^{-1}=A$.
• Consider the system of linear equations \begin{align*} x_1&= 2, \\ -2x_1 + x_2 &= 3, \\ 5x_1-4x_2 +x_3 &= 2 \end{align*} (a) Find the coefficient matrix and its inverse matrix. (b) Using the inverse matrix, solve the system of linear equations. ( The Ohio State University )
• Consider the following system of linear equations \begin{align*} 2x+3y+z&=-1\\ 3x+3y+z&=1\\ 2x+4y+z&=-2. \end{align*} (a) Find the coefficient matrix $A$ for this system. (b) Find the inverse matrix of the coefficient matrix found in (a) (c) Solve the system using the inverse matrix $A^{-1}$.
• Consider the matrix \[A=\begin{bmatrix} 1 & 2 & 1 \\ 2 &5 &4 \\ 1 & 1 & 0 \end{bmatrix}.\] (a) Calculate the inverse matrix $A^{-1}$. If you think the matrix $A$ is not invertible, then explain why. (b)  Are the vectors \[ \mathbf{A}_1=\begin{bmatrix} 1 \\ 2 \\ 1 \end{bmatrix}, \mathbf{A}_2=\begin{bmatrix} 2 \\ 5 \\ 1 \end{bmatrix}, \text{ and } \mathbf{A}_3=\begin{bmatrix} 1 \\ 4 \\ 0 \end{bmatrix}\] linearly independent? (c)  Write the vector $\mathbf{b}=\begin{bmatrix} 1 \\ 1 \\ 1 \end{bmatrix}$ as a linear combination of $\mathbf{A}_1$, $\mathbf{A}_2$, and $\mathbf{A}_3$. ( The Ohio State University, Linear Algebra Exam )
• A $2 \times 2$ matrix $A$ satisfies $\tr(A^2)=5$ and $\tr(A)=3$. Find $\det(A)$. See (a)
• Suppose that a real matrix $A$ maps each of the following vectors \[\mathbf{x}_1=\begin{bmatrix} 1 \\ 1 \\ 1 \end{bmatrix}, \mathbf{x}_2=\begin{bmatrix} 0 \\ 1 \\ 1 \end{bmatrix}, \mathbf{x}_3=\begin{bmatrix} 0 \\ 0 \\ 1 \end{bmatrix} \] into the vectors \[\mathbf{y}_1=\begin{bmatrix} 1 \\ 2 \\ 0 \end{bmatrix}, \mathbf{y}_2=\begin{bmatrix} -1 \\ 0 \\ 3 \end{bmatrix}, \mathbf{y}_3=\begin{bmatrix} 3 \\ 1 \\ 1 \end{bmatrix},\] respectively. That is, $A\mathbf{x}_i=\mathbf{y}_i$ for $i=1,2,3$. Find the matrix $A$. ( Kyoto University Exam )
• Let $A$ and $B$ are $n \times n$ matrices with real entries. Assume that $A+B$ is invertible. Then show that \[A(A+B)^{-1}B=B(A+B)^{-1}A.\] ( University of California, Berkeley Qualifying Exam )
• Let $A$ be an $n\times n$ invertible matrix. Prove that the inverse matrix of $A$ is uniques.
• Let $\mathbf{u}$ and $\mathbf{v}$ be vectors in $\R^n$, and let $I$ be the $n \times n$ identity matrix. Suppose that the inner product of $\mathbf{u}$ and $\mathbf{v}$ satisfies \[\mathbf{v}^{\trans}\mathbf{u}\neq -1.\] Define the matrix \[A=I+\mathbf{u}\mathbf{v}^{\trans}.\] Prove that $A$ is invertible and the inverse matrix is given by the formula \[A^{-1}=I-a\mathbf{u}\mathbf{v}^{\trans},\] where \[a=\frac{1}{1+\mathbf{v}^{\trans}\mathbf{u}}.\] This formula is called the Sherman-Woodberry formula .
• Let $A$ be a singular $2\times 2$ matrix such that $\tr(A)\neq -1$ and let $I$ be the $2\times 2$ identity matrix. Then prove that the inverse matrix of the matrix $I+A$ is given by the following formula: \[(I+A)^{-1}=I-\frac{1}{1+\tr(A)}A.\] Using the formula, calculate the inverse matrix of $\begin{bmatrix} 2 & 1\\ 1& 2 \end{bmatrix}$.
• Let $A=\begin{bmatrix} a & 0\\ 0& b \end{bmatrix}$. Show that (a) $A^n=\begin{bmatrix} a^n & 0\\ 0& b^n \end{bmatrix}$ for any $n \in \N$. (b) Let $B=S^{-1}AS$, where $S$ be an invertible $2 \times 2$ matrix. Show that $B^n=S^{-1}A^n S$ for any $n \in \N$
• Let $A$ be an $n\times n$ invertible matrix. Then prove the transpose $A^{\trans}$ is also invertible and that the inverse matrix of the transpose $A^{\trans}$ is the transpose of the inverse matrix $A^{-1}$. Namely, show that $(A^{\trans})^{-1}=(A^{-1})^{\trans}$.
• A square matrix $A$ is called nilpotent if there exists a positive integer $k$ such that $A^k=O$, where $O$ is the zero matrix. (a) If $A$ is a nilpotent $n \times n$ matrix and $B$ is an $n\times n$ matrix such that $AB=BA$. Show that the product $AB$ is nilpotent. (b) Let $P$ be an invertible $n \times n$ matrix and let $N$ be a nilpotent $n\times n$ matrix. Is the product $PN$ nilpotent? If so, prove it. If not, give a counterexample.
• (a) Show that if $A$ is invertible, then $A$ is nonsingular. (b) Let $A, B, C$ be $n\times n$ matrices such that $AB=C$. Prove that if either $A$ or $B$ is singular, then so is $C$. (c) Show that if $A$ is nonsingular, then $A$ is invertible.
• A square matrix $A$ is called nilpotent if some power of $A$ is the zero matrix. Namely, $A$ is nilpotent if there exists a positive integer $k$ such that $A^k=O$, where $O$ is the zero matrix. Suppose that $A$ is a nilpotent matrix and let $B$ be an invertible matrix of the same size as $A$. Is the matrix $B-A$ invertible? If so prove it. Otherwise, give a counterexample.
• Let $A$ be an $n\times n$ matrix. The $(i, j)$ cofactor $C_{ij}$ of $A$ is defined to be $C_{ij}=(-1)^{ij}\det(M_{ij})$, where $M_{ij}$ is the $(i,j)$ minor matrix obtained from $A$ removing the $i$-th row and $j$-th column. Then consider the $n\times n$ matrix $C=(C_{ij})$, and define the $n\times n$ matrix $\Adj(A)=C^{\trans}$. The matrix $\Adj(A)$ is called the adjoint matrix of $A$. When $A$ is invertible, then its inverse can be obtained by the formula \[A^{-1}=\frac{1}{\det(A)}\Adj(A).\] For each of the following matrices, determine whether it is invertible, and if so, then find the invertible matrix using the above formula. (a) $A=\begin{bmatrix} 1 & 5 & 2 \\ 0 &-1 &2 \\ 0 & 0 & 1 \end{bmatrix}$. (b) $B=\begin{bmatrix} 1 & 0 & 2 \\ 0 &1 &4 \\ 3 & 0 & 1 \end{bmatrix}$.
• Let $A$ be a real symmetric matrix whose diagonal entries are all positive real numbers. Is it true that the all of the diagonal entries of the inverse matrix $A^{-1}$ are also positive? If so, prove it. Otherwise, give a counterexample.
• Let $A$ be an $n\times n$ nonsingular matrix with integer entries. Prove that the inverse matrix $A^{-1}$ contains only integer entries if and only if $\det(A)=\pm 1$.
• Introduction to Matrices
• Elementary Row Operations
• Gaussian-Jordan Elimination
• Solutions of Systems of Linear Equations
• Linear Combination and Linear Independence
• Nonsingular Matrices
• Subspaces in $\R^n$
• Bases and Dimension of Subspaces in $\R^n$
• General Vector Spaces
• Subspaces in General Vector Spaces
• Linearly Independency of General Vectors
• Bases and Coordinate Vectors
• Dimensions of General Vector Spaces
• Linear Transformation from $\R^n$ to $\R^m$
• Linear Transformation Between Vector Spaces
• Orthogonal Bases
• Determinants of Matrices
• Computations of Determinants
• Introduction to Eigenvalues and Eigenvectors
• Eigenvectors and Eigenspaces
• Diagonalization of Matrices
• The Cayley-Hamilton Theorem
• Dot Products and Length of Vectors
• Eigenvalues and Eigenvectors of Linear Transformations
• Jordan Canonical Form

Inverse of Matrix

The inverse of Matrix for a matrix A is denoted by A -1 . The inverse of a 2 × 2 matrix can be calculated using a simple formula. Further, to find the inverse of a matrix of order 3 or higher, we need to know about the determinant and adjoint of the matrix. The inverse of a matrix is another matrix, which by multiplying with the given matrix gives the identity matrix.

The inverse of matrix is used of find the solution of linear equations through the matrix inversion method. Here, let us learn about the formula, methods, and terms related to the inverse of matrix.

What is Inverse of Matrix?

The inverse of matrix is a matrix, which on multiplication with the given matrix gives the multiplicative identity . For a square matrix A, its inverse is A -1 , and A · A -1 = A -1 · A = I, where I is the identity matrix. The matrix whose determinant is non-zero and for which the inverse matrix can be calculated is called an invertible matrix. For example, the inverse of A = \(\left[\begin{array}{rr} 1 & -1 \\ \\ 0 & 2 \end{array}\right]\) is \(\left[\begin{array}{cc} 1 & 1 / 2 \\ \\ 0 & 1 / 2 \end{array}\right]\) as

• A · A -1 = \(\left[\begin{array}{rr} 1 & -1 \\ \\ 0 & 2 \end{array}\right]\) \(\left[\begin{array}{cc} 1 & 1 / 2 \\ \\ 0 & 1 / 2 \end{array}\right]\) = \(\left[\begin{array}{cc} 1 & 0 \\ \\ 0 & 1 \end{array}\right]\) = I
• A -1 · A = \(\left[\begin{array}{cc} 1 & 1 / 2 \\ \\ 0 & 1 / 2 \end{array}\right]\) \(\left[\begin{array}{rr} 1 & -1 \\ \\ 0 & 2 \end{array}\right]\) = \(\left[\begin{array}{cc} 1 & 0 \\ \\ 0 & 1 \end{array}\right]\) = I

But how to find the inverse of a matrix? Let us see in the upcoming sections.

Inverse Matrix Formula

In the case of real numbers , the inverse of any real number a was the number a -1 , such that a times a -1 equals 1. We knew that for a real number, the inverse of the number was the reciprocal of the number, as long as the number wasn't zero. The inverse of a square matrix A, denoted by A -1 , is the matrix so that the product of A and A -1 is the identity matrix . The identity matrix that results will be the same size as matrix A.

The formula to find the inverse of a matrix is: A -1 = 1/|A| · Adj A, where

• |A| is the determinant of A and
• Adj A is the adjoint of A

Since |A| is in the denominator of the above formula, the inverse of a matrix exists only if the determinant of the matrix is a non-zero value. i.e., |A| ≠ 0.

How to Find Matrix Inverse?

To find the inverse of a square matrix A, we use the following formula: A -1 = adj(A) / |A| ; |A| ≠ 0

• A is a square matrix.
• adj(A) is the adjoint matrix of A.
• |A| is the determinant of A.

☛ Note: For a matrix to have its inverse exists:

• The given matrix should be a square matrix.
• The determinant of the matrix should not be equal to zero.

Terms Related to Matrix Inverse

The following terms below are helpful for more clear understanding and easy calculation of the inverse of matrix.

Minor: The minor is defined for every element of a matrix . The minor of a particular element is the determinant obtained after eliminating the row and column containing this element. For a matrix A = \(\begin{pmatrix} a_{11}&a_{12}&a_{13}\\a_{21}&a_{22}&a_{23}\\a_{31}&a_{32}&a_{33}\end{pmatrix}\), the minor of the element \(a_{11}\) is:

Minor of \(a_{11}\) = \(\left|\begin{matrix}a_{22}&a_{23}\\a_{32}&a_{33}\end{matrix}\right|\)

Cofactor: The cofactor of an element is calculated by multiplying the minor with -1 to the exponent of the sum of the row and column elements in order representation of that element.

Cofactor of \(a_{ij}\) = (-1) i + j × minor of \(a_{ij}\).

Determinant: The determinant of a matrix is the single unique value representation of a matrix. The determinant of the matrix can be calculated with reference to any row or column of the given matrix. The determinant of the matrix is equal to the summation of the product of the elements and its cofactors, of a particular row or column of the matrix.

Singular Matrix: A matrix having a determinant value of zero is referred to as a singular matrix . For a singular matrix A, |A| = 0. The inverse of a singular matrix does not exist.

Non-Singular Matrix: A matrix whose determinant value is not equal to zero is referred to as a non-singular matrix . For a non-singular matrix |A| ≠ 0 and hence its inverse exists.

Adjoint of Matrix: The adjoint of a matrix is the transpose of the cofactor element matrix of the given matrix.

Rules For Row and Column Operations of a Determinant:

The following rules are helpful to perform the row and column operations on determinants .

• The value of the determinant remains unchanged if the rows and columns are interchanged.
• The sign of the determinant changes, if any two rows or (two columns) are interchanged.
• If any two rows or columns of a matrix are equal, then the value of the determinant is zero.
• If every element of a particular row or column is multiplied by a constant , then the value of the determinant also gets multiplied by the constant.
• If the elements of a row or a column are expressed as a sum of elements, then the determinant can be expressed as a sum of determinants.
• If the elements of a row or column are added or subtracted with the corresponding multiples of elements of another row or column, then the value of the determinant remains unchanged.

Methods to Find Inverse of Matrix

The inverse of a matrix can be found using two methods. The inverse of a matrix can be calculated through elementary operations and through the use of an adjoint of a matrix. The elementary operations on a matrix can be performed through row or column transformations. Also, the inverse of a matrix can be calculated by applying the inverse of matrix formula through the use of the determinant and the adjoint of the matrix. For performing the inverse of the matrix through elementary column operations we use the matrix X and the second matrix B on the right-hand side of the equation.

• Elementary row or column operations
• Inverse of matrix formula (using the adjoint and determinant of matrix)

Let us check each of the methods described below.

Elementary Row Operations

To calculate the inverse of matrix A using elementary row transformations, we first take the augmented matrix [A | I], where I is the identity matrix whose order is the same as A. Then we apply the row operations to convert the left side A into I. Then the matrix gets converted into [I | A -1 ]. For a more detailed process, click here .

Elementary Column Operations

We can apply the column operations as well just like how the process was explained for row operations to find the inverse of matrix.

Inverse of Matrix Formula

The inverse of matrix A can be computed using the inverse of matrix formula, A -1 = (adj A)/(det A). i.e., by dividing the adjoint of a matrix by the determinant of the matrix. The inverse of a matrix can be calculated by following the given steps:

• Step 1: Calculate the minors of all elements of A.
• Step 2: Then compute the cofactors of all elements and write the cofactor matrix by replacing the elements of A by their corresponding cofactors.
• Step 3: Find the adjoint of A (written as adj A) by taking the transpose of the cofactor matrix of A.
• Step 4: Multiply adj A by the reciprocal of the determinant.

For a matrix A, its inverse A -1 = \(\dfrac{1}{|A|}\) · Adj A

If A = \(\begin{pmatrix} a_{11}&a_{12}&a_{13}\\a_{21}&a_{22}&a_{23}\\a_{31}&a_{32}&a_{33}\end{pmatrix}\), then

• |A| = \(a_{11}(-1)^{1 + 1} \left|\begin{matrix}a_{22}&a_{23}\\a_{32}&a_{33}\end{matrix}\right| + a_{12}(-1)^{1 + 2} \left|\begin{matrix}a_{21}&a_{23}\\a_{31}&a_{33}\end{matrix}\right| + a_{13}(-1)^{1 + 3} \left|\begin{matrix}a_{21}&a_{22}\\a_{31}&a_{32}\end{matrix}\right|\)
• Adj A = Transpose of Cofactor Matrix = Transpose of \(\begin{pmatrix} A_{11}&A_{12}&A_{13}\\A_{21}&A_{22}&A_{23}\\A_{31}&A_{32}&A_{33}\end{pmatrix}\) =\(\begin{pmatrix} A_{11}&A_{21}&A_{31}\\A_{12}&A_{22}&A_{32}\\A_{13}&A_{23}&A_{33}\end{pmatrix}\)

A -1 = \(\dfrac{1}{|A|}.\begin{pmatrix} A_{11}&A_{21}&A_{31}\\A_{12}&A_{22}&A_{32}\\A_{13}&A_{23}&A_{33}\end{pmatrix}\)

In this section, we have learned the different methods to calculate the inverse of a matrix. Let us understand it better using a few examples for the different orders of matrices in the " examples " section below.

Inverse of 2 x 2 Matrix

The inverse of 2 × 2 matrix is easier to calculate in comparison to matrices of higher order. We can calculate the inverse of 2 × 2 matrix using the general steps to calculate the inverse of a matrix. Let us find the inverse of the 2 × 2 matrix given below: A = \(\begin{bmatrix} a & b \\ \\ c & d \end{bmatrix}\) A -1 = (1/|A|) × Adj A = [1/(ad - bc)] × \(\begin{bmatrix} d & -b \\ \\ -c & a \end{bmatrix}\)

Therefore, in order to calculate the inverse of 2 × 2 matrix, we need to first swap the positions of terms a and d and put negative signs for terms b and c, and finally divide it by the determinant of the matrix.

Inverse of 3 x 3 Matrix

We know that for every non-singular square matrix A, there exists an inverse matrix A -1 , such that A × A -1 = I. Let us take any 3 × 3 square matrix given as,

A = \(\begin{bmatrix} a_{11}&a_{12}&a_{13}\\a_{21}&a_{22}&a_{23}\\a_{31}&a_{32}&a_{33}\end{bmatrix}\)

The inverse of 3x3 matrix can be calculated using the inverse matrix formula, A -1 = (1/|A|) × Adj A

We will first check if the given matrix is invertible, i.e., |A| ≠ 0. If the inverse of a matrix exists, we can find the adjoint of the given matrix and divide it by the determinant of the matrix.

A similar method can be followed to find the inverse of any n × n matrix. Let us see if similar steps can be used to calculate the inverse of m × n matrix where m ≠ n.

Inverse of 2 × 3 Matrix

We know that the first condition for the inverse of a matrix to exist is that the given matrix should be a square matrix. Also, the determinant of this square matrix should be non-zero. This means that the inverse of matrices of the order m × n will not exist where m ≠ n. Therefore, we cannot calculate the inverse of 2 × 3 matrix.

Inverse of 2 × 1 Matrix

Similar to the inverse of 2 × 3 matrix, the inverse of 2 × 1 matrix also does not exist because the given matrix is not a square matrix.

Determinant of Inverse Matrix

The determinant of the inverse of an invertible matrix is the inverse of the determinant of the original matrix. i.e., det(A -1 ) = 1 / det(A). Let us check the proof of the above statement.

We know that, det(A • B) = det (A) × det(B)

Also, A × A -1 = I

det(A × A -1 ) = det(I)

or, det(A) × det(A -1 ) = det(I)

Since, det(I) = 1

det(A) × det(A -1 ) = 1

or, det(A -1 ) = 1 / det(A)

Hence, proved.

☛ Related Articles:

• Inverse Matrix Calculator
• Matrix Multiplication Calculator
• Determinant Calculator

Important Points on Inverse of a Matrix:

• The inverse of a square matrix (if exists) is unique.
• If A and B are two invertible matrices of the same order then (AB) -1 = B -1 A -1 .
• The inverse of a square matrix A exists, only if its determinant is a non-zero value, |A| ≠ 0.
• The determinant of matrix inverse is equal to the reciprocal of the determinant of the original matrix.
• The determinant of the product of two matrices is equal to the product of the determinants of the two individual matrices. |AB| = |A|.|B|

Cuemath is one of the world's leading math learning platforms that offers LIVE 1-to-1 online math classes for grades K-12 . Our mission is to transform the way children learn math, to help them excel in school and competitive exams. Our expert tutors conduct 2 or more live classes per week, at a pace that matches the child's learning needs.

Inverse of Matrix Examples

Example 1: Find the inverse of matrix A = \(\left(\begin{matrix}-3 & 4\\ \\ 2 & 5 \end{matrix}\right)\).

The given matrix is A = \(\left(\begin{matrix}-3 & 4\\ \\2 & 5 \end{matrix}\right)\).

The formula to calculate the matrix inverse of A = \(\left(\begin{matrix}a&b\\\\c&d\end{matrix}\right)\) is A -1 = \(\dfrac{1}{ad - bc}\left(\begin{matrix}d&-b\\\\-c&a\end{matrix}\right)\).

Using this formula we can calculate A -1 as follows.

A -1 = \(\dfrac{1}{(-3)× 5 - 4 × 2}\left(\begin{matrix}5&-4\\\\-2&-3\end{matrix}\right)\)

= \(\dfrac{1}{-15 - 8}\left(\begin{matrix}5&-4\\\\-2&-3\end{matrix}\right)\)

= \(\dfrac{-1}{23}\left(\begin{matrix}5&-4\\\\-2&-3\end{matrix}\right)\)

Answer: Therefore A -1 = \(\dfrac{-1}{23}\left(\begin{matrix}5&-4\\\\-2&-3\end{matrix}\right)\)

Example 2: Find the matrix inverse of the matrix A = \(\left(\begin{matrix}4 & -2 & 1\\5&0&3\\-1&2 & 6\end{matrix}\right)\).

The given matrix is A = \(\left(\begin{matrix}4 & -2 & 1\\5&0&3\\-1&2 & 6\end{matrix}\right)\)

Step - 1: Let us find the determinant of the given matrix using Row - 1 of the above matrix.

|A| = \(4\left|\begin{matrix}0&3\\2 & 6\end{matrix}\right| -(-2)\left|\begin{matrix}5&3\\-1 & 6\end{matrix}\right|+1\left|\begin{matrix}5&0\\-1& 2\end{matrix}\right|\)

= 4(0 × 6 - 3 × 2) + 2(5 × 6 - (-1) × 3) +1(5 × 2 - 0 × (-1))

= 4(0 - 6) + 2(30 + 3) + 1(10 - 0)

= -24 + 66 + 10

Now, we will determine the adjoint of the matrix A by calculating the cofactors of each element and then taking the transpose of the cofactor matrix.

Adj A = \(\left(\begin{matrix}-6 & 14 & -6\\-33&25&-7\\10&-6 & 10\end{matrix}\right)\)

The inverse of matrix A is given by the formula A -1 = \(\dfrac{1}{|A|}\).Adj A

A -1 = \(\dfrac{1}{52}\).\(\left(\begin{matrix}-6 & 14 & -6\\-33&25&-7\\10&-6 & 10\end{matrix}\right)\)

= \(\left(\begin{matrix}-3/26 & 7/26 & -3/26\\-33/52&25/52&-7/52\\5/26&-3/26 & 5/26\end{matrix}\right)\)

Answer: A -1 = \(\left(\begin{matrix}-3/26 & 7/26 & -3/26\\-33/52&25/52&-7/52\\5/26&-3/26 & 5/26\end{matrix}\right)\)

Example 3: Find the inverse of \(\begin{bmatrix} 4 & 2 \\\\ -1 & 5 \end{bmatrix}\).

To find: Inverse of matrix \(\begin{bmatrix} 4 & 2 \\\\ -1 & 5 \end{bmatrix}\)

Using the matrix inverse formula,

\( A^{-1} = \dfrac{\text{adj(A)}}{\text{|A|}}\)

\( A^{-1} = \dfrac{1}{det \begin{pmatrix}4 & 2 \\\\ -1 & 5 \end{pmatrix}} \begin{pmatrix}5 & -2 \\\\ 1 & 4 \end{pmatrix}\)

Since, det \(\begin{pmatrix}4 & 2 \\\\ -1 & 5 \end{pmatrix}\) = 22

\( A^{-1} = \dfrac{1}{22} \begin{pmatrix}5 & -2 \\\\ 1 & 4 \end{pmatrix} = \begin{pmatrix} 5/22 & -2/22 \\\\ 1/22 & 4/22 \end{pmatrix} \)

Answer: Inverse of the given matrix \( = \begin{bmatrix} 5/22 & -1/11 \\\\ 1/22 & 2/11 \end{bmatrix}\)

go to slide go to slide go to slide

Book a Free Trial Class

Practice Questions on Inverse of Matrix

go to slide go to slide

FAQs on Inverse of Matrix

What is the meaning of inverse of matrix.

The inverse of a matrix is another matrix , which multiplies with the given matrix and gives the multiplicative identity. For a matrix A, its inverse is A -1 , and A · A -1 = I. The general formula for the inverse of matrix is equal to the adjoint of a matrix divided by the determinant of a matrix. i.e., A -1 = 1/|A| · Adj A. The inverse of a matrix exists only if the determinant of the matrix is a non-zero value.

What is the Formula for An Inverse Matrix?

The inverse matrix formula is used to determine the inverse matrix for any given matrix. The inverse of a square matrix, A is A -1 only when: A × A -1 = A -1 × A = I . The inverse matrix formula can be given as, A -1 = adj(A)/|A|; |A| ≠ 0, where A is a square matrix.

How to Calculate Inverse of Matrix?

The inverse of a square matrix is calculated in 2 simple steps.

• Step 1: First, the determinant and the adjoint of the given square matrix are calculated.
• Step 2: The inverse of the matrix A is equal to \(\dfrac{1}{|A|}\).Adj A.

How to Find Inverse of a 2 × 2 Matrix?

The inverse of a 2 × 2 matrix is equal to the adjoint of the matrix divided by the determinant of the matrix. For a matrix A = \(\left(\begin{matrix}a&b\\ \\c&d\end{matrix}\right)\), its adjoint is equal to the interchange of the elements of the first diagonal and the sign change of the elements of the second diagonal. The formula for the inverse of the matrix is as follows.

A -1 = \(\dfrac{1}{ad - bc}\left(\begin{matrix}d&-b\\\\-c&a\end{matrix}\right)\)

How to Use Inverse of Matrix?

The inverse of matrix is useful in solving equations by using the matrix inversion method. The matrix inversion method using the formula of X = A -1 B, where X is the variable matrix, A is the coefficient matrix, and B is the constant matrix.

Can Inverse of Matrix be Calculated for an Invertible Matrix?

Yes, the inverse of matrix can be calculated for an invertible matrix . The matrix whose determinant is not equal to zero is a non-singular matrix. And for a non-singular matrix, we can find the determinant and the inverse of matrix.

When Does the Matrix Inverse not Exist in Some Cases?

The inverse of matrix exists only if its determinant value is a non-zero value and when the given matrix is a square matrix. Because the adjoint of the matrix is divided by the determinant of the matrix, to obtain the inverse of a matrix. The matrix whose determinant is a non-zero value is called a non-singular matrix. Matrix inverse is not defined for rectangular matrices .

Given a 2 × 2 Matrix, What is the Formula for Finding the Inverse of the Matrix?

For a given 2×2 matrix A = \(\left(\begin{matrix}a&b\\ \\c&d\end{matrix}\right)\) , inverse is given by A -1 = \(\dfrac{1}{ad - bc}\left(\begin{matrix}d&-b\\\\-c&a\end{matrix}\right)\). Here A -1 is the inverse of A.

What is the Inverse of Identity Matrix?

The inverse of an identity matrix is itself. This is because for any identity matrix of order I, we have I × I = I × I = I. For more information, click here .

How to Use Inverse Matrix Formula?

The inverse matrix formula can be used following the given steps:

• Step 1: Find the matrix of minors for the given matrix.
• Step 2: Then find the matrix of cofactors.
• Step 3: Find the adjoint by taking the transpose of the matrix of cofactors.
• Step 4: Divide it by the determinant.

What is 3 × 3 Inverse Matrix Formula?

The matrix inverse formula for a 3 × 3 matrix is, A -1 = adj(A)/|A|; |A| ≠ 0 where A = square matrix, adj(A) = adjoint of square matrix, A -1 = inverse matrix.

What is the Inverse of Diagonal Matrix?

The inverse of a diagonal matrix is again a diagonal matrix in which the elements of the principal diagonal of the matrix inverse are the reciprocals of the corresponding elements of the original matrix. To know how to prove this, click here .

• Solve equations and inequalities
• Simplify expressions
• Factor polynomials
• Graph equations and inequalities
• Advanced solvers
• All solvers
• Arithmetics
• Determinant
• Percentages
• Scientific Notation
• Inequalities

MULTIPLICATIVE INVERSES OF MATRICES

In this section multiplicative identity elements and multiplicative inverses are introduced and used to solve matrix equations. This leads to another method for solving systems of equations. IDENTITY MATRICES The identity property for real numbers says that a * I = a and I * a = a for any real number a. If there is to be a multiplicative identity matrix I, such that:

 AI = A and  IA = A,

for any matrix A, then A and I must be square matrices of the same size. Otherwise it would not be possible to find both products. For example, let A be the 2 X 2 matrix and let

Check that with this definition of I, both Al = A and IA = A.  

Verify that MI = M and IM = M

The 2 X 2 identity matrix found above suggests the following generalization: n x n IDENTITY MATRIX

For any value of n there is an n X n identity matrix having l's down the diagonal and 0's elsewhere. The n x n identity matrix is given by l where:

Here aij = 1 when i = j (the diagonal elements) and aaj = 0 otherwise.

Example 2 - STATING AND VERIFYING THE 3 X 3 IDENTITY MATRIX Let K =

By the definition of matrix multiplication,  

Recall that l/a can also be written a^(-1). In the rest of this section, a method is developed for finding a multiplicative inverse for square matrices. The multiplicative inverse of a matrix A is written A^(-1). This matrix must satisfy the statements

The multiplicative inverse of a matrix can be found using the matrix row transformations given in the previous tutorial and repeated here for convenience.

MATRIX ROW TRANSFORMATIONS

The matrix row transformations are:

interchanging any two rows of a matrix;

multiplying the elements of any row of a matrix by the same nonzero scalar k; and

Let the unknown inverse matrix be

By the definition of matrix inverse, AA^(-1) = 1, or

Setting corresponding elements equal gives the system of equations

Since equations (1) and (3) involve only x and z, while equations (2) and (4) involve only y and w, these four equations lead to two systems of equations, 2x + 4z = 1 x-z=0

2y + 4w = 0 y-w=1. Writing the two systems as augmented matrices gives

The numbers in the first column to the right of the vertical bar give the values of x and z. The second column gives the values of y and w. That is,

To check, multiply A by A^(-1). The result should be I

Math Topics

More solvers.

• Add Fractions
• Simplify Fractions

Want Better Math Grades?

✅ Unlimited Solutions

✅ Step-by-Step Answers

✅ Available 24/7

➕ Free Bonuses ($1085 value!)

On this page

• Matrices and Determinants
• 1. Determinants
• Systems of 3x3 Equations interactive applet
• 2. Large Determinants
• 3. Matrices
• 4. Multiplication of Matrices
• 4a. Matrix Multiplication examples
• 4b. Add & multiply matrices applet
• 5. Finding the Inverse of a Matrix
• 5a. Simple Matrix Calculator
• 5b. Inverse of a Matrix using Gauss-Jordan Elimination
• 6. Matrices and Linear Equations
• Matrices and Linear Transformations
• Eigenvalues and eigenvectors - concept applet
• 7. Eigenvalues and Eigenvectors
• 8. Applications of Eigenvalues and Eigenvectors
• Eigenvalues and eigenvectors calculator

Related Sections

Math Tutoring

Need help? Chat with a tutor anytime, 24/7.

Online Algebra Solver

Solve your algebra problem step by step!

IntMath Forum

Get help with your math queries:

Inverse of a Matrix using Gauss-Jordan Elimination

by M. Bourne

In this section we see how Gauss-Jordan Elimination works using examples.

You can re-load this page as many times as you like and get a new set of numbers each time. You can also choose a different size matrix (at the bottom of the page).

(If you need some background first, go back to the Introduction to Matrices ).

Choose the matrix size you are interested in and then click the button.

The randomly-generated example appears below.

Phone users

NOTE: If you're on a phone, you can scroll any wide matrices on this page to the right or left to see the whole expression.

Example (3 × 3)

Find the inverse of the matrix A using Gauss-Jordan elimination.

Our Procedure

We write matrix A on the left and the Identity matrix I on its right separated with a dotted line, as follows. The result is called an augmented matrix.

We include row numbers to make it clearer.

Next we do several row operations on the 2 matrices and our aim is to end up with the identity matrix on the left , like this:

(Technically, we are reducing matrix A to reduced row echelon form , also called row canonical form ).

The resulting matrix on the right will be the inverse matrix of A .

Our row operations procedure is as follows:

• We get a "1" in the top left corner by dividing the first row
• Then we get "0" in the rest of the first column
• Then we need to get "1" in the second row, second column
• Then we make all the other entries in the second column "0".

We keep going like this until we are left with the identity matrix on the left.

Let's now go ahead and find the inverse.

We start with:

New Row [1]

Divide Row [1] by 9 (to give us a "1" in the desired position):

This gives us:

New Row [2]

Row[2] − 13 × Row[1] (to give us 0 in the desired position):

13 − 13 × 1 = 0 7 − 13 × 0.8889 = -4.5556 4 − 13 × 0.5556 = -3.2222 0 − 13 × 0.1111 = -1.4444 1 − 13 × 0 = 1 0 − 13 × 0 = 0

This gives us our new Row [2]:

New Row [3]

Row[3] − 6 × Row[1] (to give us 0 in the desired position):

6 − 6 × 1 = 0 3 − 6 × 0.8889 = -2.3333 14 − 6 × 0.5556 = 10.667 0 − 6 × 0.1111 = -0.6667 0 − 6 × 0 = 0 1 − 6 × 0 = 1

This gives us our new Row [3]:

Divide Row [2] by -4.5556 (to give us a "1" in the desired position):

Row[1] − 0.8889 × Row[2] (to give us 0 in the desired position):

1 − 0.8889 × 0 = 1 0.8889 − 0.8889 × 1 = 0 0.5556 − 0.8889 × 0.7073 = -0.0732 0.1111 − 0.8889 × 0.3171 = -0.1707 0 − 0.8889 × -0.2195 = 0.1951 0 − 0.8889 × 0 = 0

This gives us our new Row [1]:

Row[3] − -2.3333 × Row[2] (to give us 0 in the desired position):

0 − -2.3333 × 0 = 0 -2.3333 − -2.3333 × 1 = 0 10.667 − -2.3333 × 0.7073 = 12.317 -0.6667 − -2.3333 × 0.3171 = 0.0732 0 − -2.3333 × -0.2195 = -0.5122 1 − -2.3333 × 0 = 1

Divide Row [3] by 12.317 (to give us a "1" in the desired position):

Row[1] − -0.0732 × Row[3] (to give us 0 in the desired position):

1 − -0.0732 × 0 = 1 0 − -0.0732 × 0 = 0 -0.0732 − -0.0732 × 1 = 0 -0.1707 − -0.0732 × 0.0059 = -0.1703 0.1951 − -0.0732 × -0.0416 = 0.1921 0 − -0.0732 × 0.0812 = 0.0059

Row[2] − 0.7073 × Row[3] (to give us 0 in the desired position):

0 − 0.7073 × 0 = 0 1 − 0.7073 × 0 = 1 0.7073 − 0.7073 × 1 = 0 0.3171 − 0.7073 × 0.0059 = 0.3129 -0.2195 − 0.7073 × -0.0416 = -0.1901 0 − 0.7073 × 0.0812 = -0.0574

We have achieved our goal of producing the Identity matrix on the left. So we can conclude the inverse of the matrix A is the right hand portion of the augmented matrix:

Things to Note

• The above explanation shows all steps. A human can usually take a few shortcuts. Also, sometimes there is already a "1" or a "0" in the correct position, and in those cases, we would not need to do anything for that step.
• Always write down what you are doing in each step - it is very easy to get lost!
• I have shown results correct to 4 decimal place, but best possible accuracy was used throughout. Be aware that small errors from rounding will accumulate throughout the problem. Always use full calculator accuracy! (Make full use of your calculator's memory.)
• Very occasionally there are strange results because of the computer's internal representation of numbers. That is, it may store "1" as 0.999999999872.

See another?

You can go back up to the top of the page and choose another example.

Tips, tricks, lessons, and tutoring to help reduce test anxiety and move to the top of the class.

Email Address Sign Up

• HW Guidelines
• Study Skills Quiz
• Find Local Tutors
• Demo MathHelp.com
• Join MathHelp.com

Select a Course Below

• ACCUPLACER Math
• Math Placement Test
• PRAXIS Math
• + more tests
• 5th Grade Math
• 6th Grade Math
• Pre-Algebra
• College Pre-Algebra
• Introductory Algebra
• Intermediate Algebra
• College Algebra

How to Find the Inverse of a Matrix

Intro A Warning & A Word Problem

Can you divide by a matrix?

For matrices, there is no such thing as division. You can add, subtract , and multiply matrices, but you cannot divide them. There is a related concept, though, which is called "inversion". First I'll discuss why inversion is useful, and then I'll show you how to do it.

Content Continues Below

MathHelp.com

Advertisement

This terminology and these facts are very important for matrices. If you are given a matrix equation like AX  =  C , where you are given A and C and are told to figure out X , you would like to divide off the matrix A . But you can't do division with matrices.

What is an inverse matrix, in simple words?

In simple terms, an inverse matrix is the square matrix A −1 that you can multiply on either side of matrix A to get the identity matrix I . In other words, given matrix A , its inverse matrix A −1 obeys the following:

A × A −1 = A −1 × A = I

Why would you want to invert a matrix? What are they for?

Given the matrix equation AX  =  C , what if you could find the inverse of A , something similar to finding reciprocal fractions for solving linear equations? You can use the inverse of A , written as A −1 and pronounced " A inverse", to cancel off the A from the matrix equation; this then allows you then to solve the matrix equation for X .

A −1 AX = A −1 C

IX = A −1 C

X = A −1 C

Inversion works the same way for matrices. If you multiply a matrix (such as A ) and its inverse (in this case, A −1 ), you get the identity matrix I , which is the matrix analog of the number 1 . And the point of the identity matrix is that IX  =  X for any matrix X (meaning "any matrix of the correct size", of course).

It should be noted that the order in the multiplication above is important and is not at all arbitrary. Recall that, for matrices, multiplication is not commutative. That is, AB is almost never equal to BA .

So multiplying the matrix equation "on the left" (to get A −1 AX ) is not at all the same thing as multiplying "on the right" (to get AXA −1 ). The product AXA −1 does not equal A −1 AX , because you can't switch around the order in matrix multiplication.

Thus, to solve the matrix equation AX=C , you have to multiply A −1 on the left, putting it right next to the A in the original matrix equation. And since you have to do the same thing to both sides of an equation when you're solving, you must multiply "on the left" on the right-hand side of the equation as well, resulting in A −1 C .

You cannot be casual with your placement of the matrices; you must be precise, correct, and consistent. This is the only way to successfully cancel off A and solve the matrix equation.

As you have seen above, inverse matrices can be very useful for solving matrix equations. But—

How do you find the inverse of a matrix?

To find the inverse of a matrix, follow these steps:

• Write out the matrix that you're wanting to invert.
• Append to this matrix the identity matrix, making one matrix that is now twice as wide as it is tall.
• Using row operations , convert the left-hand half of the double-wide in the identity matrix.
• The new right-hand side of the double-wide is the inverse of the original matrix.

This technique for inverting matrices is kind of clever. Here's an example of how it works:

First, I write down the entries the matrix A , but I write them in a double-wide matrix:

In the other half of the double-wide, I write the identity matrix:

Now I'll do matrix row operations to convert the left-hand side of the double-wide into the identity. (As always with row operations, there is no one "right" way to do this. What follows are just the steps that happened to occur to me. Your calculations could easily look quite different.)

Now that the left-hand side of the double-wide contains the identity, the right-hand side contains the inverse. That is, the inverse matrix is the following:

How can you know that this matrix is the inverse?

Note that we can confirm that this matrix is the inverse of A by multiplying the two matrices and seeing that we get the identity.

Since the multiplication ended with the identity matrix, the matrix we found is confirmed to be the inverse of the original matrix that they gave us.

Be advised that, in "real life", the inverse is rarely a matrix filled with nice neat whole numbers like this. With any luck, though, especially if you're doing inverses by hand, you'll be given nice ones like this to do.

Is there a formula for inverting a 2x2 matrix?

To find the inverse of a 2-by-2 matrix, use the following formula:

For the following matrix:

...the inverse matrix is given by:

Is there a formula for inverting a 3x3 matrix?

There is a formula, sort of , for the inverse of a 3-by-3 matrix, but it's arguably not the quickest way to proceed. Use the method above instead.

Are there other ways to find the inverse of a matrix?

There are loads of ways to find the inverse of a matrix; Wikipedia gives an extensive list ( link ). Following the swap-the-identity-matrix method above is probably the easiest of the listed methods. But, in real life (as a mathematician or scientist), other solution methods are preferred. *Any* method is preferred to inverse matrices.

URL: https://www.purplemath.com/modules/mtrxinvr.htm

You can use the Mathway widget below to practice finding the inverse of a matrix. Try the entered exercise, or type in your own exercise. Then click the button and select "Find the Inverse" to compare your answer to Mathway's. (Or skip the widget and continue with the lesson.)

Please accept "preferences" cookies in order to enable this widget.

(Click "Tap to view steps" to be taken directly to the Mathway site for a paid upgrade.)

Page 1 Page 2

Standardized Test Prep

College math, homeschool math, share this page.

• Terms of Use
• Privacy / Cookies
• About Purplemath
• About the Author
• Tutoring from PM
• Advertising
• Linking to PM
• Site licencing

Visit Our Profiles

Inverse of a matrix

Table of Contents

What is the inverse of a matrix?

When can a matrix be inverted, how to find the inverse of a 2×2 matrix.

As you can see, inverting a 2×2 dimension matrix is simple: you only have to solve the determinant of the matrix (|A|), switch the elements on the main diagonal, and change the sign of the elements on the secondary diagonal.

First, we take the determinant of the 2×2 matrix:

Now we apply the formula of the inverse matrix:

Invert the following 2×2 dimension matrix:

The determinant of matrix A is:

How to find the inverse of a 3×3 matrix

The determinant is nonzero, therefore, the matrix can be inverted.

We transpose the cofactor matrix to find the adjugate matrix:

Properties of the inverse matrix

The inverse matrix has the following characteristics:

Solving a system of equations with the inverse matrix

We can verify that these matrices correspond to the system of equations by multiplying the matrices, since we would obtain the two equations of the system.

Now, to simplify the following steps, we will name A the square matrix that has the coefficients of the unknowns, X the column matrix with the unknowns, and B the column matrix with the independent terms:

So matrix X is the unknown of the matrix equation.

1 thought on “Inverse of a matrix”

Leave a comment cancel reply.

• Unit Conversions
• Sets and Types of Numbers
• Common Multiple and Divisor
• Fractions and Decimals
• Algebraic Expressions and Polynomials
• Equations and Inequalities
• Sequences and Series
• Combinatorics
• Probability and Statistics
• Stereometry
• Analytical Geometry
• Sum, Difference and Product of Matrices
• Inverse Matrix
• Rank of a Matrix
• Determinant of a Matrix
• Matrix Equations
• System of Equations Solved by Matrices
• Matrix Word Problems
• Limits, Derivatives, Integrals
• Analysis of Functions
• Trig table and chart

to the matrix :               Link Partners Privacy Policy Copyright © 2015-2023 math-exercises.com - All rights reserved. Any use of website content without written permission is prohibited. Math Lessons Math Formulas Calculators Math Calculators, Lessons and Formulas It is time to solve your math problem HW Help (paid service) Solving System of Linear Equations Inverse Matrix Method Cramers Rule Suppose you are given an equation in one variable such as $4x = 10$. Then you will find the value of $x$ that solves this equation by multiplying the equation by the inverse of 4: $\color{blue}{\frac14} \cdot 4x = \color{blue}{\frac14} \cdot 10$, so the solution will be $x = 2.5$. Sometimes we can do something very similar to solve systems of linear equations; in this case, we will use the inverse of the coefficient matrix. But first we must check that this inverse exists! The conditions for the existence of the inverse of the coefficient matrix are the same as those for using Cramer's rule, that is 1. The system must have the same number of equations as variables, that is, the coefficient matrix of the system must be square. 2. The determinant of the coefficient matrix must be non-zero. The reason, of course, is that the inverse of a matrix exists precisely when its determinant is non-zero. 3. To use this method follow the steps demonstrated on the following system: Step 1: Rewrite the system using matrix multiplication: and writing the coefficient matrix as A, we have Step 2: FInd the inverse of the coefficient matrix A. In this case the inverse is Step 3: Multiply both sides of the equation (that you wrote in step #1) by the matrix $A^{-1}$. On the left you'll get On the right, you get and so the solution is Roots and Radicals Simplify Expression Adding and Subtracting Multiplying and Dividing Complex Numbers Arithmetic Polar representation Polynomials Multiplying Polynomials Division of Polynomials Zeros of Polynomials Rational Expressions Simplifying Multiplying and Dividing Adding and Subtracting Solving Equations Linear Equations Absolute Value Equations Quadratic Equation Equations with Radicals Systems of Equations Substitution Method Elimination Method Row Reduction Cramers Rule Inverse Matrix Method Exponential Functions Introduction Exponential Equations Logarithmic Functions Trigonometry Trigonometric Formulas Trigonometric Equations Law of Cosines Progressions Arithmetic Progressions Geometric Progressions Differentiation Common formulas Product and Quotient Rule Chain Rule Limits Properties of Limits Rational Function Irrational Functions Trigonometric Functions L'Hospital's Rule Integrals Integration Formulas Exercises Integral techniques Substitution Integration by Parts Integrals with Trig. Functions Trigonometric Substitutions Integral Applications Area Volume Arc Length Analytic geometry Analytic Geometry 2D Basic Concepts Lines Parallel and Perpendicular Lines Polar Coordinates Conic Sections Circle Ellipse Hyperbola Analytic Geometry 3D Line in 3D Planes Linear Algebra Matrices Definitions Addition and Multiplication Gauss-Jordan elimination Determinants Introduction to Determinants Applications of Determinants Vectors Basic Operations Dot Product Cross Product Random Quote If people do not believe that mathematics is simple, it is only because they do not realize how complicated life is. John Louis von Neumann It is a mathematical fact that fifty percent of all doctors graduate in the bottom half of their class. Author Unknown Welcome to MathPortal. This website's owner is mathematician Miloš Petrović. I designed this website and wrote all the calculators, lessons, and formulas . If you want to contact me, probably have some questions, write me using the contact form or email me on [email protected] Email (optional) Online Matrix Inverse Calculator Invert matrices with wolfram|alpha. Natural Language More than just an online matrix inverse calculator Wolfram|Alpha is the perfect site for computing the inverse of matrices. Use Wolfram|Alpha for viewing step-by-step methods and computing eigenvalues, eigenvectors, diagonalization and many other properties of square and non-square matrices. Learn more about: Matrices, eigenvectors and eigenvalues Tips for entering queries Use plain English or common mathematical syntax to enter your queries. To enter a matrix, separate elements with commas and rows with curly braces, brackets or parentheses. inv{{2,3},{4,7}} Inverse{{1,2,3},{4,5,6},{7,8,9}} find the inverse of the matrix ((a,3),(5,-7)) {{2/3,-5/7}, {-3,4/9}}^-1 inverse of [[2,3],[5,6]] inverse of [[1,2],[3,6]] View more examples Access instant learning tools Get immediate feedback and guidance with step-by-step solutions and Wolfram Problem Generator Step-by-step solutions Wolfram Problem Generator School Guide Mathematics Number System and Arithmetic Trigonometry Probability Mensuration Maths Formulas Class 8 Maths Notes Class 9 Maths Notes Class 10 Maths Notes Class 11 Maths Notes Class 12 Maths Notes Inverse of a Matrix The inverse of Matrix is the matrix that on multiplying with the original matrix results in an identity matrix. For any matrix A, its inverse is denoted as A -1 . Let’s learn about the Matrix Inverse in detail, including its definition, formula, methods on how to find the inverse of a matrix, and examples. Table of Content Matrix Inverse Terms related to matrix inverse, how to find inverse of matrix. Inverse of a Matrix Formula Inverse Matrix Method Inverse of 2×2 matrix example, determinant of inverse matrix, properties of inverse of matrix, matrix inverse solved examples. The inverse of a matrix is another matrix that, when multiplied by the given matrix, yields the multiplicative identity . For matrix A and its inverse of A -1 , the identity property holds. A.A -1 = A -1 A = I where I is the identity matrix. The terminology listed below can help you grasp the inverse of a matrix more clearly and easily. Terms The minor of an element in a matrix is the determinant of the matrix formed by removing the row and column of that element. For element a  , remove the ith row and jth column to form a new matrix and find its determinant. Minor of is the determinant of [Tex]A = \begin{bmatrix}5 & 6\\ 6 & 7\end{bmatrix} [/Tex] The cofactor of an element is the minor of that element multiplied by , where i and j are the row and column indices of the element. Cofactor of a = (-1) Minor of a   Cofactor of = × Minor of = Minor of The determinant of a matrix is calculated as the sum of the products of the elements of any row or column and their respective cofactors. For a row (or column), sum up the product of each element and its cofactor. Determinant of A = ​× Cofactor of + × Cofactor of ​ + × Cofactor of ​. The adjoint of a matrix is the transpose of its cofactor matrix. Create a matrix of cofactors for each element of the original matrix and then transpose it. Adjoint of A is the transpose of the matrix formed by the cofactors of all elements in A. Singular Matrix A matrix whose value of the determinant is zero is called a singular matrix , i.e. any matrix A is called a singular matrix if |A| = 0. Inverse of a singular matrix does not exist. Non-Singular Matrix A matrix whose value of the determinant is non-zero is called a non-singular matrix, i.e. any matrix A is called a non-singular matrix if |A| ≠ 0. Inverse of a non-singular matrix exists. Identity Matrix A square matrix in which all the elements are zero except for the principal diagonal elements is called the identity matrix. It is represented using I. It is the identity element of the matrix as for any matrix A, An example of an Identity matrix is, I 3×3 = [Tex] \begin{bmatrix} 1 & 0 & 0\\ 0 & 1 & 0 \\ 0 & 0 & 1 \\\end{bmatrix}[/Tex] This is an identity matrix of order 3×3.\ Read More : There are Two-ways to find the Inverse of a matrix in mathematics: Using Matrix Formula Using Inverse Matrix Methods The inverse of matrix A, that is A -1 is calculated using the inverse of matrix formula, which involves dividing the adjoint of a matrix by its determinant. [Tex]A^{-1}=\frac{\text{Adj A}}{|A|}   [/Tex] adj A = adjoint of the matrix A, and  |A| = determinant of the matrix A. Note : This formula only works on Square matrices. To find inverse of matrix using inverse of a matrix formula, follow these steps. Step 1: Determine the minors of all A elements. Step 2: Next, compute the cofactors of all elements and build the cofactor matrix by substituting the elements of A with their respective cofactors. Step 3: Take the transpose of A’s cofactor matrix to find its adjoint (written as adj A). Step 4: Multiply adj A by the reciprocal of the determinant of A. Now, for any non-singular square matrix A, A -1 = 1 / |A| × Adj (A) Example: Find the inverse of the matrix  [Tex]A=\left[\begin{array}{ccc}4 & 3 & 8\\6 & 2 & 5\\1 & 5 & 9\end{array}\right][/Tex]  using the formula. We have,  [Tex]A=\left[\begin{array}{ccc}4 & 3 & 8\\6 & 2 & 5\\1 & 5 & 9\end{array}\right] [/Tex] Find the adjoint of matrix A by computing the cofactors of each element and then getting the cofactor matrix’s transpose. adj A =  [Tex]\left[\begin{array}{ccc}-7 & -49 & 28\\13 & 28 & -17\\-1 & 28 & -10\end{array}\right] [/Tex] Find the value of determinant of the matrix. |A| = 4(18–25) – 3(54–5) + 8(30–2) ⇒ |A| = 49 So, the inverse of the matrix is, A –1 =  [Tex]\frac{1}{49}\left[\begin{array}{ccc}-7 & -49 & 28\\13 & 28 & -17\\-1 & 28 & -10\end{array}\right] [/Tex] ⇒ A –1 =  [Tex]\left[\begin{array}{ccc}- \frac{1}{7} & \frac{13}{49} & – \frac{1}{49}\\-1 & \frac{4}{7} & \frac{4}{7}\\\frac{4}{7} & – \frac{17}{49} & – \frac{10}{49}\end{array}\right] [/Tex] There are two Inverse matrix methods to find matrix inverse: Determinant Method Elementary Transformation Method Method 1: Determinant Method The most important method for finding the matrix inverse is using a determinant. The inverse matrix is also found using the following equation: A -1 = adj(A) / det(A) adj(A) is the adjoint of a matrix A, and det(A) is the determinant of a matrix A. For finding the adjoint of a matrix A the cofactor matrix of A is required. Then adjoint (A) is the transpose of the Cofactor matrix of A i.e., adj (A) = [C ij ] T For the cofactor of a matrix i.e., C ij , we can use the following formula: C ij = (-1) i+j det (M ij ) where M ij refers to the (i, j) th minor matrix when i th row and j th column is removed. Method 2: Elementary Transformation Method Follow the steps below to find an Inverse matrix by elementary transformation method. Step 1 : Write the given matrix as A = IA, where I is the identity matrix of the order same as A. Step 2 : Use the sequence of either row operations or column operations till the identity matrix is achieved on the LHS also use similar elementary operations on the RHS such that we get I = BA. Thus, the matrix B on RHS is the inverse of matrix A. Step 3 : Make sure we either use Row Operation or Column Operation while performing elementary operations. We can easily find the inverse of the 2 × 2 Matrix using the elementary operation. Let’s understand this with the help of an example. Example: Find the inverse of the 2 × 2, A =   [Tex]\begin{bmatrix}2 & 1\\ 1 & 2\end{bmatrix}[/Tex]  using the elementary operation. Given: A = IA [Tex]\begin{bmatrix}2 & 1\\ 1 & 2\end{bmatrix}~=~\begin{bmatrix}1 & 0\\ 0 & 1\end{bmatrix}~×~\begin{bmatrix}2 & 1\\ 1 & 2\end{bmatrix}[/Tex] Now, R 1 ⇢ R 1 /2 [Tex]\begin{bmatrix}1 & 1/2\\ 1 & 2\end{bmatrix}~=~\begin{bmatrix}1/2 & 0\\ 0 & 1\end{bmatrix}~×~A [/Tex] R 2 ⇢ R 2 – R 1 [Tex]\begin{bmatrix}1 & 1/2\\ 0 & 3/2\end{bmatrix}~=~\begin{bmatrix}1/2 & 0\\ -1/2 & 1\end{bmatrix}~×~A [/Tex] R 2 ⇢ R 2 × 2/3 [Tex]\begin{bmatrix}1 & 1/2\\ 0 & 1\end{bmatrix}~=~\begin{bmatrix}1/2 & 0\\-1/3 & 2/3\end{bmatrix}~×~A [/Tex] R 1 ⇢ R 1 – R 2 /2 [Tex]\begin{bmatrix}1 & 0\\ 0 & 1\end{bmatrix}~=~\begin{bmatrix}2/3 & -1/6\\ -1/3 & 2/3\end{bmatrix}~×~A [/Tex] Thus, the inverse of the matrix A =  [Tex] \begin{bmatrix}2 & 1\\ 1 & 2\end{bmatrix}   [/Tex]  is A -1 =  [Tex]\begin{bmatrix}2/3 & -1/6\\ -1/3 & 2/3\end{bmatrix} [/Tex] Inverse of the 2×2 matrix can also be calculated using the shortcut method apart from the method discussed above. Let’s consider an example to understand the shortcut method to calculate the inverse of 2 × 2 Matrix. For given matrix A =  [Tex]\begin{bmatrix}a & b\\ c & d\end{bmatrix} [/Tex] We know, |A| = (ad – bc) and adj A =  [Tex]\begin{bmatrix}d & -b\\ -c & a\end{bmatrix} [/Tex] then using the formula for inverse A -1 =  (1 / |A|) × Adj A ⇒ A -1 =  [Tex][1 / (ad – bc)] × \begin{bmatrix}d & -b\\ -c & a\end{bmatrix} [/Tex] Thus, the inverse of the 2 × 2 matrix is calculated. Inverse of 3X3 Matrix Example Let us take any 3×3 Matrix A =  [Tex]\begin{bmatrix}a & b & c\\ l & m & n\\ p & q & r\end{bmatrix} [/Tex] The inverse of 3×3 matrix is calculated using the inverse matrix formula ,  A -1 = (1 / |A|) × Adj A Determinant of inverse matrix is the reciprocal of the determinant of the original matrix. i.e.,  det(A -1 ) = 1 / det(A) The proof of the above statement is discussed below: det(A × B) = det (A) × det(B)  (already know) ⇒ A × A -1 = I  (by Inverse matrix property) ⇒ det(A × A -1 ) = det(I) ⇒ det(A) × det(A -1 ) = det(I)     [ but, det(I) = 1] ⇒ det(A) × det(A -1 ) = 1 ⇒ det(A -1 ) = 1 / det(A) Hence, Proved. Inverse matrix has the following properties: For any non-singular matrix A, (A -1 ) -1 = A For any two non-singular matrices A and B, (AB) -1 = B -1 A -1 Inverse of a non-singular matrix exists, for a singular matrix, the inverse does not exist. For any nonsingular A, (A T ) -1 = (A -1 ) T Invertible Matrix Matrices: Properties and Formulas Mathematical Operation on Matrices Determinant of Matrix How to find the Determinant of Matrix? Let’s solve some example questions on Inverse of Matrix. Example 1: Find the inverse of the matrix  [Tex]\bold{A=\left[\begin{array}{ccc}2 & 3 & 1\\1 & 1 & 2\\2 & 3 & 4\end{array}\right]}[/Tex]  using the formula. We have, [Tex]A=\left[\begin{array}{ccc}2 & 3 & 1\\1 & 1 & 2\\2 & 3 & 4\end{array}\right] [/Tex] Find the adjoint of matrix A by computing the cofactors of each element and then getting the cofactor matrix’s transpose. adj A =  [Tex]\left[\begin{array}{ccc}-2 & -9 & 5\\0 & 6 & -3\\1 & 0 & -1\end{array}\right] [/Tex] Find the value of determinant of the matrix. |A| = 2(4–6) – 3(4–4) + 1(3–2) = –3 So, the inverse of the matrix is, A –1 =  [Tex]\frac{1}{-3}\left[\begin{array}{ccc}-2 & -9 & 5\\0 & 6 & -3\\1 & 0 & -1\end{array}\right] [/Tex]       =  [Tex]\left[\begin{array}{ccc}\frac{2}{3} & 3 & – \frac{5}{3}\\0 & -2 & 1\\- \frac{1}{3} & 0 & \frac{1}{3}\end{array}\right] [/Tex] Example 2: Find the inverse of the matrix A=\bold{ using the formula.} [Tex]\left[\begin{array}{ccc}6 & 2 & 3\\0 & 0 & 4\\2 & 0 & 0\end{array}\right]       [/Tex] We have, A= [Tex]\left[\begin{array}{ccc}6 & 2 & 3\\0 & 0 & 4\\2 & 0 & 0\end{array}\right] [/Tex] Find the adjoint of matrix A by computing the cofactors of each element and then getting the cofactor matrix’s transpose. adj A =  [Tex]\left[\begin{array}{ccc}0 & 0 & 8\\8 & -6 & -24\\0 & 4 & 0\end{array}\right] [/Tex] Find the value of determinant of the matrix. |A| = 6(0–4) – 2(0–8) + 3(0–0)      = 16 So, the inverse of the matrix is, A –1 =  [Tex]\frac{1}{16}\left[\begin{array}{ccc}0 & 0 & 8\\8 & -6 & -24\\0 & 4 & 0\end{array}\right] [/Tex]       =  [Tex]\left[\begin{array}{ccc}0 & 0 & \frac{1}{2}\\\frac{1}{2} & – \frac{3}{8} & – \frac{3}{2}\\0 & \frac{1}{4} & 0\end{array}\right] [/Tex] Example 3: Find the inverse of the matrix A= [Tex]\bold{\left[\begin{array}{ccc}1 & 2 & 3\\0 & 1 & 4\\0 & 0 & 1\end{array}\right]      }      [/Tex]  using the formula. We have, A= [Tex]\left[\begin{array}{ccc}1 & 2 & 3\\0 & 1 & 4\\0 & 0 & 1\end{array}\right] [/Tex] Find the adjoint of matrix A by computing the cofactors of each element and then getting the cofactor matrix’s transpose. adj A =  [Tex]\left[\begin{array}{ccc}1 & -2 & 5\\0 & 1 & -4\\0 & 0 & 1\end{array}\right] [/Tex] Find the value of determinant of the matrix. |A| = 1(1–0) – 2(0–0) + 3(0–0) = 1 So, the inverse of the matrix is, A –1 =  [Tex]\frac{1}{1}\left[\begin{array}{ccc}1 & -2 & 5\\0 & 1 & -4\\0 & 0 & 1\end{array}\right] [/Tex] =  [Tex]\left[\begin{array}{ccc}1 & -2 & 5\\0 & 1 & -4\\0 & 0 & 1\end{array}\right] [/Tex] Example 4: Find the inverse of the matrix A= [Tex]\bold{\left[\begin{array}{ccc}1 & 2 & 3\\2 & 1 & 4\\3 & 4 & 1\end{array}\right]      }      [/Tex]  using the formula. We have, A= [Tex]\left[\begin{array}{ccc}1 & 2 & 3\\2 & 1 & 4\\3 & 4 & 1\end{array}\right] [/Tex] Find the adjoint of matrix A by computing the cofactors of each element and then getting the cofactor matrix’s transpose. adj A =  [Tex]\left[\begin{array}{ccc}-15 & 10 & 5\\10 & -8 & 2\\5 & 2 & -3\end{array}\right] [/Tex] Find the value of determinant of the matrix. |A| = 1(1–16) – 2(2–12) + 3(8–3) = 20 So, the inverse of the matrix is, A –1 =  [Tex]\frac{1}{20}\left[\begin{array}{ccc}-15 & 10 & 5\\10 & -8 & 2\\5 & 2 & -3\end{array}\right] [/Tex]       =  [Tex]\left[\begin{array}{ccc}- \frac{3}{4} & \frac{1}{2} & \frac{1}{4}\\\frac{1}{2} & – \frac{2}{5} & \frac{1}{10}\\\frac{1}{4} & \frac{1}{10} & – \frac{3}{20}\end{array}\right] [/Tex] Frequently Asked Questions on Inverse of Matrix What is inverse of matrix. Reciprocal of a matrix is called the Inverse of a matrix. Only square matrices with non-zero determinants are invertible. Suppose for any square matrix A with inverse matrix B their product is always an identity matrix (I) of the same order. [A]×[B] = [I] What is Matrix? Matrix is a rectangular array of numbers that are divided into a defined number of rows and columns. The number of rows and columns in a matrix is referred to as its dimension or order. What is the Inverse of 2×2 Matrix? For any matrix A or order 3×3 its inverse is found using the formula, A -1 = (1 / |A|) × Adj A What is the Inverse of 3×3 Matrix? The inverse of any square 3×3 matrix (say A) is the matrix of the same order denoted by A -1 such that their product is an Identity matrix of order 3×3. [A] 3×3 × [A -1 ] 3×3 = [I] 3×3 Are Adjoint and Inverse of Matrix the same? No, the adjoint of a matrix and the inverse of a matrix are not the same.  How to use the Inverse of Matrix? The inverse of a matrix is used for solving algebraic expressions in matrix form. For example, to solve AX = B, where A is the coefficient matrix, X is the variable matrix and B is the constant matrix. Here the variable matrix is found using the inverse operation as, X = A -1 B What are Invertible Matrices? The matrices whose inverse exist are called invertible. Invertible matrices are matrices that have a non-zero determinant. Why does Inverse of 2 × 3 Matrix not exist? The inverse of only a square matrix exists. As the 2 × 3 matrix is not a square matrix but rather a rectangular matrix thus, its inverse does not exist. Similarly, the 2 × 1 matrix is also not a square matrix but rather a rectangular matrix thus, its inverse does not exist. What is Inverse of Identity Matrix? The inverse of an identity matrix is the identity matrix itself. This is because the identity matrix, denoted as I (or I n for an n × n matrix), is the only matrix for which every element along the main diagonal is 1 and all other elements are 0. When we multiply an identity matrix by itself (or its inverse), we get the identity matrix again. Please Login to comment... Similar reads. School Learning Maths-Class-12 Improve your Coding Skills with Practice Number Line \begin{pmatrix}1 & 2 \\3 & 4\end{pmatrix}^{-1} inverse\:\begin{pmatrix}1 & 2 & 3 \\4 & 5 & 6 \\7 & 2 & 9\end{pmatrix} \begin{pmatrix}1 & 3 & 5 & 9 \\1 & 3 & 1 & 7 \\4 & 3 & 9 & 7 \\5 & 2 & 0 & 9\end{pmatrix}^{-1} matrix-inverse-calculator The Matrix, Inverse For matrices there is no such thing as division, you can multiply but can’t divide. Multiplying by the inverse... We want your feedback Please add a message. Message received. Thanks for the feedback. Solving the wide-band inverse scattering problem via equivariant neural networks New citation alert added. This alert has been successfully added and will be sent to: You will be notified whenever a record that you have chosen has been cited. To manage your alert preferences, click on the button below. New Citation Alert! Please log in to your account Information & Contributors Bibliometrics & citations, view options, recommendations, solution of inverse problems using multilayer quaternion neural networks. Neural network models extended to higher-dimensional numbers have been studied in recent years. In particular, quaternions have an advantage with respect to the expression of rotation in a three-dimensional space. On the other hand, the problem that ... Inverted L-shape Planer Band Notch Monopole Antenna For Ultra Wide Band Communication Applications In this paper, Inverted L-Shape radiating patch with rectangular ground planer notch band monopole antenna is proposed. Antenna is fabricated on FR4 substrate with permittivity 4.4 and loss tangent 0.02 with dimension 12x8x1.6 mm3. Measured return loss ... Dual-band band pass filter using stub-loaded open-loop resonators with wide controllable bandwidths Two dual-band band pass filters BPF using stub-loaded open-loop SLOL resonator are presented in this article. A novel coupling tuning method by changing the relative coupling position of the resonators is proposed to control the bandwidth of each ... Information Published in. Elsevier Science Publishers B. V. Netherlands Publication History Author tags. Inverse scattering Inverse problems Neural networks Equivariance, Butterfly factorization Research-article Contributors Other metrics, bibliometrics, article metrics. 0 Total Citations 0 Total Downloads Downloads (Last 12 months) 0 Downloads (Last 6 weeks) 0 View options Login options. Check if you have access through your login credentials or your institution to get full access on this article. Full Access Share this publication link. Copying failed. Share on social media Affiliations, export citations. Please download or close your previous search result export first before starting a new bulk export. Preview is not available. By clicking download, a status dialog will open to start the export process. The process may take a few minutes but once it finishes a file will be downloadable from your browser. You may continue to browse the DL while the export process is in progress. Download Download citation Copy citation We are preparing your search results for download ... We will inform you here when the file is ready. Your file of search results citations is now ready. Your search export query has expired. Please try again. On an inverse problem of determining electromagnetic parameters in Maxwell’s equations from partial boundary measurements Published: 05 August 2024 Volume 75 , article number  159 , ( 2024 ) Cite this article Christian Daveau 1 , Islem Ben Hnia 2 & Abdessatar Khelifi 3   Explore all metrics In this paper, we deal with an inverse boundary value problem for the Maxwell equations with boundary data assumed known only in accessible part \(\Gamma \) of the boundary. We aim to prove uniqueness results using the Dirichlet to Neumann data with measurements limited to an open part of the boundary and we seek to reconstruct the complex refractive index \({{\varvec{n}}}\) in the interior of a body. Further, using the impedance map restricted to \(\Gamma \) , we may identify locations of small volume fraction perturbations of the refractive index. This is a preview of subscription content, log in via an institution to check access. Access this article Subscribe and save. Get 10 units per month Download Article/Chapter or eBook 1 Unit = 1 Article or 1 Chapter Cancel anytime Price includes VAT (Russian Federation) Instant access to the full article PDF. Rent this article via DeepDyve Institutional subscriptions Similar content being viewed by others On the Determination of a Coefficient of an Elliptic Equation via Partial Boundary Measurement Simultaneous determination of two coefficients in the Riemannian hyperbolic equation from boundary measurements On the solution of waveguide boundary value problems in the absence of the lorentz calibration, data availibility. No datasets were generated or analyzed during the current study. Alessandrini, G.: Stable determination of conductivity by boundary measurements. Appl. Anal. 27 , 153–172 (1988) Article   MathSciNet   Google Scholar   Ammari, H., Uhlmann, G.: Reconstruction of the potential from partial Cauchy data for the Schrodinger equation. Indiana Univ. Math. J. 53 (1), 169–184 (2004) Ammari, H., Vogelius, M., Volkov, D.: Asymptotic formulas for perturbations in the electromagnetic fields due to the presence of imperfections of small diameter II. The full Maxwell equations. J. Math. Pures Appl. 80 , 769–814 (2001) Brown, B.M., Marletta, M., Reyes, J.M.: Uniqueness for an inverse problem in electromagnetism with partial data. J. Differ. Equ. 260 (8), 6525–6547 (2016) Buffa, A., Costabel, M., Sheen, D.: On traces for \(H({curl},\Omega )\) in Lipschitz domains. J. Math. Anal. Appl. 276 , 845–867 (2002) Bukhgeim, A.L., Uhlmann, G.: Recovering a potential from partial Cauchy data. Comm. PDE 27 , 653–668 (2002) Cakoni, F., Colton, D., Gintides, D.: The interior transmission eigen-value problem. SIAM J. Math. Anal. 42 (6), 2912–2921 (2010) Calderón, A.P.: On an inverse boundary value problem. In: Seminar on Numerical Analysis and its Applications to Continuum Physics (Río de Janeiro, 1980), pp. 65–73. Soc. Brasil. Mat., Río de Janeiro (1980) Caro, P., Zhou, T.: Global uniqueness for an IBVP for the time-harmonic Maxwell equations. Anal. PDE 7 (2), 375–405 (2014) Caro, P., Ola, P., Salo, M.: Inverse boundary value problem for Maxwell equations with local data. Commun. Partial Differ. Equ. 34 , 1425–1464 (2009) Chesnel, L.: Interior transmission eigenvalue problem for Maxwell’s equations: the T-coercivity as an alternative approach. Inverse Prob. 28 , 065005 (2012) Colton, D.L., Kress, R.: Integral Equation Methods in Scattering Theory. Pure and Applied Mathematics. A Wiley-Interscience Publication, John Wiley & Sons Inc., New York (1983) Google Scholar   Colton, D., Päivärinta, L.: The uniqueness of a solution to an inverse scattering problem for electromagnetic waves. Arch. Rational Mech. Anal. 119 , 59–70 (1992) Darbas, M., Lohrengel, S.: Numerical reconstruction of small perturbations in the electromagnetic coefficients of a dielectric material. J. Comput. Math. 32 (1), 21–38 (2014) Daveau, C., Khelifi, A.: On the perturbation of the electromagnetic energy due to the presence of small inhomogeneities. C. R. Math. Acad. Sci. Paris 346 (5–6), 287–292 (2008) Daveau, C., Khelifi, A., Sushchenko, A.: Reconstruction of closely spaced small inhomogeneities via boundary measurements for the full time-dependent Maxwell’s equations. Appl. Math. Model. 33 , 1719–1728 (2009) Isakov, V.: On uniqueness in the inverse conductivity problem with local data. Inverse Probl. Imaging 1 (1), 95–105 (2007) Joshi, M., McDowall, S.R.: Total determination of material parameters from electromagnetic boundary information. Pac. J. Math. 193 , 107–129 (2000) Kenig, C.E., Sjöstrand, J., Uhlmann, G.: The Calderón problem with partial data. Ann. Math. 165 , 567–591 (2007) Kenig, C.E., Salo, M., Uhlmann, G.: Inverse problems for the anisotropic Maxwell equations. Duke Math. J. 157 , 369–419 (2011) Kohn, R., Vogelius, M.: Determining conductivity by boundary measurements. Commun. Pure Appl. Math. 37 , 289–298 (1984) Kurylev, Y., Lassas, M.: Inverse problems and index formulae for Dirac operators. Adv. Math. 221 , 170–216 (2009) Kurylev, Y., Lassas, M., Somersalo, E.: Maxwell’s equations with a polarization independent wave velocity: direct and inverse problems. J. Math. Pures Appl. 86 , 237–270 (2006) Liu, H., Yamamoto, M., Zou, J.: Reflection principle for the Maxwell equations and its application to inverse electro-magnetic scattering. Inverse Probl. 23 , 2357–2366 (2007) Article   Google Scholar   McDowall, S.R.: Boundary determination of material parameters from electromagnetic boundary information. Inverse Probl. 13 , 153–163 (1997) McDowall, S.R.: An electromagnetic inverse problem in chiral media. Trans. Am. Math. Soc. 352 (7), 2993–3013 (2000) Nachmann, A.: Reconstructions from boundary measurements. Ann. Math. 128 , 531–587 (1988) Nédélec, J.-C.: Acoustic and Electromagnetic Equations. Integral Representations for Harmonic Problems. Vol. 144 of Applied Mathematical Sciences, Springer-Verlag, New York (2001) Book   Google Scholar   Ola, P., Somersalo, E.: Electromagnetic inverse problems and generalized Sommerfeld potentials. SIAM J. Appl. Math. 56 (4), 1129–1145 (1996) Ola, P., Päivärinta, L., Somersalo, E.: An inverse boundary value problem in electrodynamics. Duke Math. J. 70 (3), 617–653 (1993) Ola, P., Päivärinta, L., Somersalo, E.: Inverse Problems for Time Harmonic Electrodynamics. Inside Out: Inverse Problems and Applications. Math. Sci. Res. Inst. Publ., vol. 47, pp. 169–191. Cambridge University Press, Cambridge (2003) Romanov, V.G., Kabanikhin, S.I.: Inverse Problems for Maxwell’s Equations. VSP, Utrecht (1994) Somersalo, E., Isaacson, D., Cheney, M.: A linearized inverse boundary value problem for Maxwell’s equations. J. Comput. Appl. Math. 42 , 123–136 (1992) Sun, Z., Uhlmann, G.: An inverse boundary value problem for Maxwell’s equations. Arch. Rational Mech. Anal. 119 , 71–93 (1992) Sylvester, J., Uhlmann, G.: A global uniqueness theorem for an inverse boundary value problem. Ann. Math. 125 , 153–169 (1987) Uhlmann, G., Wang, J.N.: Reconstruction of discontinuities using complex geometrical optics solutions. SIAM J. Appl. Math. 68 , 1026–1044 (2008) Yamamoto, M.: A mathematical aspect of inverse problems for non-stationary Maxwell’s equations. Int. J. Appl. Electromag. Mech. 8 , 77–98 (1997) Yamamoto, M.: On an inverse problem of determining source terms in Maxwell’s equations with a single measurement. Inverse Probl. Tomogr. Image Process. 15 , 241–256 (1998) MathSciNet   Google Scholar   Download references Author information Authors and affiliations. Laboratoire AGM (CNRS UMR 8088) Department of Mathematics, CY Cergy Paris université, 2 Avenue Adolphe Chauvin, 95302, Cergy-Pontoise Cedex, France Christian Daveau CY Cergy Paris université, 2 Avenue Adolphe Chauvin, 95302, Cergy-Pontoise Cedex, France Islem Ben Hnia Département de Mathématiques, Université de Carthage, Bizerte, Tunisia Abdessatar Khelifi You can also search for this author in PubMed   Google Scholar Contributions A.K proposed the main idea of the work, C.D and A.K wrote the main manuscript text after drawing together the schematic way of this investigation, and I.B reviewed the main proofs. Corresponding author Correspondence to Abdessatar Khelifi . Ethics declarations Conflict of interest. The authors declare no Conflict of interest. Additional information Publisher's note. Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. Rights and permissions Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law. Reprints and permissions About this article Daveau, C., Ben Hnia, I. & Khelifi, A. On an inverse problem of determining electromagnetic parameters in Maxwell’s equations from partial boundary measurements. Z. Angew. Math. Phys. 75 , 159 (2024). https://doi.org/10.1007/s00033-024-02299-4 Download citation Received : 14 March 2024 Revised : 18 July 2024 Accepted : 19 July 2024 Published : 05 August 2024 DOI : https://doi.org/10.1007/s00033-024-02299-4 Share this article Anyone you share the following link with will be able to read this content: Sorry, a shareable link is not currently available for this article. Provided by the Springer Nature SharedIt content-sharing initiative Inverse problem Maxwell equations Electromagnetic parameters Partial data Reconstruction Mathematics Subject Classification Find a journal Publish with us Track your research share this! August 5, 2024 This article has been reviewed according to Science X's editorial process and policies . Editors have highlighted the following attributes while ensuring the content's credibility: fact-checked peer-reviewed publication trusted source Optimization for inverse problem solving in computer-generated holography by Chinese Academy of Sciences Computer-generated holography (CGH) provides an approach to digitally modulate a given wavefront. This technology, partly inherited from optical holography and partly advanced by the progress of computing technology, has become an emerging focus of academia and industry. Computer-generated holograms, encoded on various types of holographic media, enable a wide range of applications. Holograms fabricated as diffractive optical elements or metasurfaces can reproduce specific spatial light fields, achieving structured light projection, data storage, and optical encryption. With refreshable devices like spatial light modulators, CGH is able to assist many fields of investigation, including three-dimensional (3D) display, holographic lithography, optical trapping, and optogenetics. In recent years, CGH has also boosted the birth and growth of potential markets of virtual reality (VR), augmented reality (AR), head-up display (HUD), and optical computing. Although the applications and fields of investigation involve various elements and devices, the algorithms for hologram synthesis can be universally applied. In a new paper published in Light: Science & Applications , a team of scientists, led by Professor Liangcai Cao from the Department of Precision Instruments, Tsinghua University, China, and Professor Daping Chu from the Centre for Photonic Devices and Sensors, University of Cambridge, UK review the optimization algorithms applied to computer-generated holography, incorporating principles of hologram synthesis based on alternative projections and gradient descent methods. Finding the exact solution of a desired hologram to reconstruct an accurate target object constitutes an ill-posed inverse problem. The general practice of single-diffraction computation for synthesizing holograms can only provide an approximate answer and it is subject to limitations in numerical implementation. Various non-convex optimization algorithms are thus designed to seek an optimal solution by introducing different constraints, frameworks, and initializations. "Optimization algorithms generate holograms by solving the inverse problem in CGH, but simply optimization does not necessarily guarantee the generation of an appropriate hologram. Careful calculation corresponding to the actual physics process is essential to improve the reconstructing accuracy and produce a hologram as it should be, hence the importance and value of this best practice guide," explained the researchers. "This review article is focused on the development and implementation of CGH optimization algorithms in real operation, providing a systematical and comprehensive overview from fundamental to actual practice. "We believe together with our open-source codes which demonstrate all the 2D/3D optimization frameworks presented here, this computation tutorial as well as the incorporated understanding of optimization algorithms for CGH can assist various investigations in the general field of optics" they added. Journal information: Light: Science & Applications Provided by Chinese Academy of Sciences Explore further Feedback to editors How do butterflies stick to branches during metamorphosis? Historic fires trapped in Antarctic ice yield key information for climate models Hubble spotlights a supernova 2 hours ago New technology uses light to engrave erasable 3D images Seagrasses filter human pathogens in marine waters 3D laser printing with bioinks from microalgae 3 hours ago PhAI—an AI system that figures out the phase of x-rays that crystals have diffracted Testing the viability of using horse milk to make ice cream 4 hours ago Study reveals how the Global North drives inequality in international trade Pioneering plasma-catalytic process for CO₂ hydrogenation to methanol under ambient conditions 5 hours ago Relevant PhysicsForums posts The double-slit experiment with a pit in the screen. 23 hours ago How to observe difference of L/R rotation intensity of CPL Jul 29, 2024 Increase in numerical aperture leads to a decrease in line width? Jul 19, 2024 Ray Tracing Software for Broadband Lasers Jul 18, 2024 The detection of the carrier-envelope offset frequency (fCEO) of optical signal Jul 16, 2024 Looking for information on Verity SD100 monochromator unit Jul 12, 2024 More from Optics Related Stories Diffractive optical networks reconstruct holograms instantaneously without a digital computer Nov 2, 2021 Advancing real-time 3D holographic display: A new method for computer-generated holography Apr 8, 2024 Technology advance paves way to more realistic 3D holograms for virtual reality and more Apr 6, 2023 Using model-driven deep learning to achieve high-fidelity 4K color holographic display Feb 22, 2023 Liquid lens–based holographic camera captures real 3D scenes Mar 6, 2024 Simplifying the generation of three-dimensional holographic displays Oct 18, 2023 Recommended for you X-ray imagery of vibrating diamond opens avenues for quantum sensing Aug 7, 2024 Glimpse into the nanoworld: Microscope reveals tiniest cell processes Hidden harmonies: Team discovers magnon–phonon Fermi resonance in an antiferromagnet Lasers deliver powerful shocking punch in material experiments Coupling excitons to polaritons for better solar cells and higher intensity LEDs Researchers develop energy-efficient optical neural networks Aug 6, 2024 Let us know if there is a problem with our content Use this form if you have come across a typo, inaccuracy or would like to send an edit request for the content on this page. For general inquiries, please use our contact form . For general feedback, use the public comments section below (please adhere to guidelines ). Please select the most appropriate category to facilitate processing of your request Thank you for taking time to provide your feedback to the editors. Your feedback is important to us. However, we do not guarantee individual replies due to the high volume of messages. E-mail the story Your email address is used only to let the recipient know who sent the email. Neither your address nor the recipient's address will be used for any other purpose. The information you enter will appear in your e-mail message and is not retained by Phys.org in any form. Newsletter sign up Get weekly and/or daily updates delivered to your inbox. You can unsubscribe at any time and we'll never share your details to third parties. More information Privacy policy Donate and enjoy an ad-free experience We keep our content available to everyone. Consider supporting Science X's mission by getting a premium account. E-mail newsletter 433 COMMENTS Inverse Matrix Questions Inverse matrix questions and solutions are given here to help students learn how to find the inverse of different matrices using different formulas and techniques. As we know, matrices are one of the most scoring concepts for students. Finding the inverse matrix is simple for 2×2 matrices. However, we can easily find the inverse matrix for 3× ... 2.4.1: Inverse Matrices (Exercises) SECTION 2.4 PROBLEM SET: INVERSE MATRICES In problems 5 - 6, find the inverse of each matrix by the row-reduction method. Inverse of a Matrix Also called the Gauss-Jordan method. This is a fun way to find the Inverse of a Matrix: Play around with the rows (adding, multiplying or swapping) until we make Matrix A into the Identity Matrix I. And by ALSO doing the changes to an Identity Matrix it magically turns into the Inverse! The "Elementary Row Operations" are simple things like ... 2.7: Finding the Inverse of a Matrix How can we find the inverse of a matrix, if it exists? This section introduces the method of Gaussian elimination and the concept of elementary matrices to answer this question. You will also learn how to check if a matrix is invertible and how to use its inverse to solve systems of linear equations. This section is part of a first course in linear algebra, which covers the basics of matrices ... Inverse of a Matrix Inverse of a Matrix. We write A-1 instead of 1 A because we don't divide by a matrix! And there are other similarities: When we multiply a number by its reciprocal we get 1: 8 × 1 8 = 1. When we multiply a matrix by its inverse we get the Identity Matrix (which is like "1" for matrices): A × A -1 = I. Inverse Matrix Inverse matrix can be calculated using different methods. Learn what is inverse matrix, how to find the inverse matrix for 2x2 and 3x3 matrices along with the steps and solved examples here at BYJU'S. Inverse Matrix Questions with Solutions Inverse Matrix Questions with Solutions Tutorials including examples and questions with detailed solutions on how to find the inverse of square matrices using the method of the row echelon form and the method of cofactors. The properties of inverse matrices are discussed and various questions, including some challenging ones, related to inverse matrices are included along with their detailed ... Inverse Matrices Problems of Inverse Matrices. From introductory exercise problems to linear algebra exam problems from various universities. Basic to advanced level. PDF Introduction to Linear Algebra, 5th Edition 2.5 Inverse Matrices ' If the square matrix A has an inverse, then both A−1A = I and AA−1 = I. The algorithm to test invertibility is elimination: A must have n (nonzero) pivots. The algebra test for invertibility is the determinant of A : det A must not be zero. 7.6: Inverse Matrices In this section, we will learn to find the inverse of a matrix, if it exists. Later, we will use matrix inverses to solve linear systems. 5. Finding the Inverse of a Matrix We learn how to find the inverse of a matrix, whihc we use later to solve systems of linear equations. Inverse of Matrix The inverse of matrix is used of find the solution of linear equations through the matrix inversion method. Here, let us learn about the formula, methods, and terms related to the inverse of matrix. Find matrix inverse with Step-by-Step Math Problem Solver In this section multiplicative identity elements and multiplicative inverses are introduced and used to solve matrix equations. This leads to another method for solving systems of equations. Inverse of a Matrix using Gauss-Jordan Elimination A step-by-step explanation of finding the inverse of a matrix using Gauss-Jordan Elimination. Up to 5x5 matrix. What is an inverse matrix, and what's it used for? The inverse of a matrix A is A⁻¹, just as the inverse of 2 is ½. We can solve equations by multiplying through by inverses; it's similar with matrices. How to find the inverse of a matrix (formula and examples) Example. We are going to calculate the inverse of the following 2×2 square matrix: First, we take the determinant of the 2×2 matrix: Now we apply the formula of the inverse matrix: And we multiply the matrix by the fraction: So the inverse of matrix A is: As you can see, inverting a matrix with this formula is very fast, but it can only be ... Math Exercises & Math Problems: Inverse Matrix High school, college and university math exercises on inverse matrix, inverse matrices. Find the inverse matrix to the given matrix at Math-Exercises.com. 3.5: Matrix Inverses Learn about invertible transformations, and understand the relationship between invertible matrices and invertible transformations. Recipes: compute the inverse matrix, solve a linear system by taking inverses. Picture: the inverse of a transformation. Vocabulary words: inverse matrix, inverse transformation. Inverse Matrix Method Inverse Matrix Method Suppose you are given an equation in one variable such as . Then you will find the value of that solves this equation by multiplying the equation by the inverse of 4: , so the solution will be . Online Matrix Inverse Calculator Access instant learning tools. Get immediate feedback and guidance with step-by-step solutions and Wolfram Problem Generator. Free online inverse matrix calculator computes the inverse of a 2x2, 3x3 or higher-order square matrix. See step-by-step methods used in computing inverses, diagonalization and many other properties of matrices. Inverse of a Matrix The inverse matrix is denoted by A-1. The inverse of a matrix is a special matrix that gives an identity matrix by multiplying it with the original matrix. Matrix Inverse Calculator Free matrix inverse calculator - calculate matrix inverse step-by-step. Solving the wide-band inverse scattering problem via equivariant neural This paper introduces a novel deep neural network architecture for solving the inverse scattering problem in frequency domain with wide-band data, by directly approximating the inverse map, thus avoiding the expensive optimization loop of classical methods. On an inverse problem of determining electromagnetic ... In this paper, we deal with an inverse boundary value problem for the Maxwell equations with boundary data assumed known only in accessible part $$\\Gamma $$ Γ of the boundary. We aim to prove uniqueness results using the Dirichlet to Neumann data with measurements limited to an open part of the boundary and we seek to reconstruct the complex refractive index $${{\\varvec{n}}}$$ n in the ... 7.7: Solving Systems with Inverses What is the best method to solve this problem? There are several ways we can solve this problem. As we have seen in previous sections, systems of equations and matrices are useful in solving real-world problems involving finance. After studying this section, we will have the tools to solve the bond problem using the inverse of a matrix. Optimization for inverse problem solving in computer-generated holography a, The synthesis of computer-generated holograms can be described as an inverse problem. b, Constraints, frameworks, and initialization need to be considered in hologram optimization. c ... arXiv:2408.02199v1 [math.NA] 5 Aug 2024 A Fast Fourier-Galerkin Method for Solving Boundary Integral Equations on Torus-Shaped Surfaces ... the decay pattern of the entries in the representation matrix. Leveraging this decay pattern, we devise a truncation strat- ... which also possesses a differentiable inverse. The interior Dirichlet problem involves finding a functionu satisfying ... essay homework paper phd project resume review speech study thesis