Finding the inverse of a matrix can be tricky but once understood properly, you can tackle any question. The multiplication of a matrix by its inverse is equal to the identity matrix. It means that when you multiply the original matrix with the inverse of that matrix, it will always result in the identity matrix.

A . { A }^{ -1 } = { A }^{ -1 } . A = I

For example, you are asked to find the inverse of 15. What you should do is reciprocate the number because inverse means negative exponent({ 15 }^{ -1 }), to make the exponent positive, you need to reciprocate the number({ 15 }^{ -1 } = \frac { 1 }{ 15 }). We do the same for matrix, let's say we have a matrix A, now the inverse of A will be { A }^{ -1 }, however, how will you reciprocate the whole matrix? It isn't easy, that is why we have a method for finding the inverse of a matrix and it is called the Gaussian elimination method. Gaussian elimination method is not the only method to find the inverse of a matrix, there are a few more methods but this method is the easiest and that is why, in this lesson, we will cover how to use the Gaussian elimination method to find the inverse of a matrix.

Properties of the Inverse Matrix

There are 4 properties of the inverse matrix which are listed below:

1. If a matrix is nonsingular then its inverse matrix will also be nonsingular

{({ A }^{ -1 })}^{ -1 } = A

2. If two nonsingular matrices are multiplied with each other they will still produce a nonsingular matrix

{ (AB) }^{ -1 } = { B }^{ -1 } { A }^{ -1 }

3. If a matrix is nonsingular then { ({ A }^{ T }) }^{ -1 } = { ({ A }^{ -1 }) }^{ T }

4. Suppose two matrics, A and B, with their product resulting in identity matrix, AB = I, we can say that A and B matrices are inverses of each other.

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Steps to Calculate the Inverse Matrix

Gaussian Elimination Method

A is a square matrix of order n. To calculate the inverse of A, denoted as { A }^{ -1 }, follow these steps:

Step 1:

Construct a matrix of type M = (A | I), that is to say, A is in the left half of M and the identity matrix I is on the right.

Consider an arbitrary 3x3 matrix:

A = \begin{pmatrix} 1 & 1 & 0 \\ 1 & 0 & 1 \\ 0 & 1 & 0 \end{pmatrix}

Place the identity matrix of order 3 to the right of Matrix M like below:

\begin{pmatrix} 1 & 1 & 0 & : & 1 & 0 & 0 \\ 1 & 0 & 1 & : & 0 & 1 & 0 \\ 0 & 1 & 0 & : & 0 & 0 & 1 \end{pmatrix}

Step 2:

Using the Gaussian elimination method, transform the left half, A, to the identity matrix, located to the right, and the matrix that results in the right side will be the inverse of the matrix: { A }^{ -1 }.

{ r }_{ 2 } \rightarrow { r }_{ 2 } - { r }_{ 1 }

\begin{pmatrix} 1 & 1 & 0 & : & 1 & 0 & 0 \\ 0 & -1 & 1 & : & -1 & 1 & 0 \\ 0 & 1 & 0 & : & 0 & 0 & 1 \end{pmatrix}

 

{ r }_{ 3 } \rightarrow { r }_{ 3 } + { r }_{ 2 }

\begin{pmatrix} 1 & 1 & 0 & : & 1 & 0 & 0 \\ 0 & -1 & 1 & : & -1 & 1 & 0 \\ 0 & 0 & 1 & : & -1 & 1 & 1 \end{pmatrix}

 

{ r }_{ 2 } \rightarrow { r }_{ 2 } - { r }_{ 3 }

\begin{pmatrix} 1 & 1 & 0 & : & 1 & 0 & 0 \\ 0 & -1 & 0 & : & 0 & 0 & -1 \\ 0 & 0 & 1 & : & -1 & 1 & 1 \end{pmatrix}

 

{ r }_{ 1 } \rightarrow { r }_{ 1 } + { r }_{ 2 }

\begin{pmatrix} 1 & 0 & 0 & : & 1 & 0 & -1 \\ 0 & -1 & 0 & : & 0 & 0 & -1 \\ 0 & 0 & 1 & : & -1 & 1 & 1 \end{pmatrix}

 

(-1) . { r }_{ 2 }

\begin{pmatrix} 1 & 0 & 0 & : & 1 & 0 & -1 \\ 0 & 1 & 0 & : & 0 & 0 & 1 \\ 0 & 0 & 1 & : & -1 & 1 & 1 \end{pmatrix}

The inverse matrix is:

{ A }^{ -1 } = \begin{pmatrix} 1 & 0 & -1 \\ 0 & 0 & 1 \\ -1 & 1 & 1 \end{pmatrix}

 

Example

Calculate the matrix inverse of:

A = \begin{pmatrix} 1 & -1 & 0 \\ 0 & 1 & 0 \\ 2 & 0 & 1 \end{pmatrix}

Construct a matrix of type M = (A | I).

\begin{pmatrix} 1 & -1 & 0 & : & 1 & 0 & 0 \\ 0 & 1 & 0 & : & 0 & 1 & 0 \\ 2 & 0 & 1 & : & 0 & 0 & 1 \end{pmatrix}

Using the Gaussian elimination method, transform the left half, A, in the identity matrix, located to the right, and the matrix that results in the right side will be the inverse of matrix : { A }^{ -1 }.

 

r3 - 2r1 r3 - 2r2
\begin{pmatrix} 1 & -1 & 0 & : & 1 & 0 & 0 \\ 0 & 1 & 0 & : & 0 & 1 & 0 \\ 0 & 2 & 1 & : & -2 & 0 & 1 \end{pmatrix} \begin{pmatrix} 1 & -1 & 0 & : & 1 & 0 & 0 \\ 0 & 1 & 0 & : & 0 & 1 & 0 \\ 0 & 0 & 1 & : & -2 & -2 & 1 \end{pmatrix}
r1 + r2 Matrix Inverse
\begin{pmatrix} 1 & 0 & 0 & : & 1 & 1 & 0 \\ 0 & 1 & 0 & : & 0 & 1 & 0 \\ 0 & 0 & 1 & : & -2 & -2 & 1 \end{pmatrix}  { A }^{ -1 } = \begin{pmatrix} 1 & 1 & 0 \\ 0 & 1 & 0 \\ -2 & -2 & 1 \end{pmatrix}

 

For what values of m in the matrix A = \begin{pmatrix} 1 & 1 & m \\ m & 0 & -1 \\ 6 & -1 & 0 \end{pmatrix} does not support an inverse?

\left| A \right| = \begin{vmatrix} 1 & 1 & m \\ m & 0 & -1 \\ 6 & -1 & 0 \end{vmatrix} = 0 - { m }^{ 2 } - 6 - 0 - 1 - 0 = -{ m }^{ 2 } - 7

-{ m }^{ 2 } - 7 = 0 \qquad m = \pm \sqrt { -7 } \notin R

For any real value of m, there is the inverse { A }^{ -1 }A-1.

 

Calculating the matrix inverse for determinants

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Emma

I am passionate about travelling and currently live and work in Paris. I like to spend my time reading, gardening, running, learning languages and exploring new places.