The identity matrix I_{ n} is the square matrix with order n x n and with the elements in the main diagonal consiting of 1's and all other elements are equal to zero.

Examples:

The 2 x 2 identity matrix

I_{ 2} = [1 0]

[0 1]

The 3 x 3 identity matrix

[1 0 0]

[0 1 0]

[0 0 1]

In general the n x n matrix identity has the form

[1 0 ... 0]

[0 1 ... 0]

[0 0 ... 0]

A = [. ]

[. ]

[0 0 ... 1]

## Definition of The Inverse of a Matrix

Let A be a square matrix of orde n x n. If there exists a matrix B such that

A B = I _{ n} = B A

Then B is called the inverse matrix of A. (and of course matrix A is the inverse matrix of B).

Examples:

Verify that matrices A and B given below are inverses of each other.

A = [2 3]

[3 4]

B = [-4 3]

[3 -2]

Solution

Let us find the products AB and BA

AB = [2 3][-4 3] = [-8 + 9 6 - 6] = [1 0]

[3 4][3 -2] [-12 + 12 9 - 8] [0 1]

BA = [-4 3][2 3] = [-8 + 9 -12 + 12] = [1 0]

[3 -2][3 4] [6 - 6 9 - 8] [0 1]

We have verified that AB = BA = I_{ 2} and therefore A and B are inverse of each other.

The inverse of a square matrix A is denoted as A^{ -1}.

## How to Find The Inverse of a 2 by 2 Square Matrix?

__Example 1:__ Find a formula for the inverse of a 2 x 2 square matrix A given by

A = [a b]
[c d]

__Solution__

Let B be the inverse of A given by

B = [a' b']
[c' d']

From definition, we have

AB = I

[a b][a' b'] = [1 0]

[c d][c' d'] [0 1]

We need to find terms a', b', c' and d' of B (inverse) in terms of a, b, c and d which are the terms of matrix A. Multiply matrices A and B above.

[a a' + b c' a b' + b d']= [1 0]

[c a' + d c' c b' + d d'] [0 1]

The above gives 4 equations

a a' + b c' = 1 (equation 1)

c a' + d c' = 0 (equation 2)

a b' + b d' = 0 (equation 3)

c b' + d d' = 1 (equation 4)

We now need to solve the first and second equations simultaneously to find a' and c' in terms of a, b, c and d.

Multiply the first equation by d and the second equation by b and subtract the left and right terms of the equations obtained to find

a'(a d - b c) = d

a ' = d / (a d - b c)

Substitute a' in the second equation to obtain c'= - c / (a d - b c)

To find b' and d', multiply equation (3) by d and equation (4) by b and subtract the left and right terms of the equations obtained to find

b'(a d - b c) = - b

or b' = - b / (a d - b c)

Substitute b' in equation (3) to obtain d'= a / (a d - b c)

Finally matrix B , the inverse of A, is given by

B = 1 / (a d - b c) [d -b]

[-c a]

In solving the question in the above example, we have found a formula for the inverse of any invertible 2 x 2 matrix. A 2 x 2 matrix will have an inverse if it determinant D = a d - b c is not equal to zero since division by zero is not allowed.

__Example 2:__ Use the above formula for the inverse of a 2 x 2 square matrix to find the inverse of

A = [3 5]
[2 3]

__Solution__

Acoording to formula above the inverse of A , A^{ -1} is given by

A^{ -1} = 1 / [ (3)(3) - (5)(2) ] [ 3 -5 ]

[-2 3]

Simplify to obtain
A^{ -1} = [ -3 5 ]

[2 -3]

As an exercise, show that the product A A^{ -1} = A^{ -1} A = I_{ 2}.

## How to Find The Inverse of a 3 by 3 Square Matrix?

__Example 1:__ Find the inverse of a 3 x 3 square matrix A given by

[1 1 2]

A = [2 4 -3]

[3 6 -5]

__Solution__

We use a method based on augmenting the given matrix by the identity matrix as follows

[1 1 2 | 1 0 0] row(1)

[2 4 -3 | 0 1 0] row(2)

[3 6 -5 | 0 0 1] row(3)

We next use basic row operations so that the identity matrix on the right is moved to the left

We multiply row(1) by - 2 and add it to row(2) and put the result back in row(2). We also multiply row(1) by - 3 and add it to row(3) and put the result back in row(3).

[1 1 2 | 1 0 0] row(1)

[0 2 -7 | -2 1 0] row(2)

[0 3 -11 | -3 0 1] row(3)

We multiply row(2) by 3 and row(3) by -2, add them and put the row obtained in row(3).

[1 1 2 | 1 0 0] row(1)

[0 2 -7 | -2 1 0] row(2)

[0 0 1 | 0 3 -2] row(3)

We multiply row(1) by -2 and add it to row(2) and put the row obtained in row(1).
[-2 0 -11 | -4 1 0] row(1)

[0 2 -7 | -2 1 0] row(2)

[0 0 1 | 0 3 -2] row(3)

We multiply row(3) by 11 and add it to row(1) and put the row obtained in row(1).
[-2 0 0 | -4 34 -22] row(1)

[0 2 -7 | -2 1 0] row(2)

[0 0 1 | 0 3 -2] row(3)

We multiply row(3) by 7 and add it to row(2) and put the row obtained in row(2).
[-2 0 0 | -4 34 -22] row(1)

[0 2 0 | -2 22 -14] row(2)

[0 0 1 | 0 3 -2] row(3)

We now multiply all terms in row(1) by -1/2 and multiply all terms in row(2) by 1/2.
[1 0 0 | 2 -17 11] row(1)

[0 1 0 | -1 11 -7] row(2)

[0 0 1 | 0 3 -2] row(3)

Now that we have the identity matrix on the left, the 3 by 3 matrix on the right is the inverse matrix. The inverse of A is given by
[2 -17 11]

A^{ -1} = [-1 11 -7]

[0 3 -2]

A similar method based on row operations can in principle be used to find the inverse of any invertible (that has an inverse) square matrix.
__Exercises__

Find the inverse of matrices A and B given below.

A = [2 3]

[3 5]

[1 1 2]

B = [1 2 4]

[2 -1 0]

__Answers__

A^{ -1} = [5 -3]

[-3 2]

[2 -1 0]

B^{ -1} = [4 -2 -1]

[-5/2 3/2 1/2]

More pages and references related to matrices.
Find Inverse Matrix - Calculator.

Find Inverse of 3 by 3 Matrix - Calculator.

Matrix Addition and Multiplication.

multiplication of matrices using an applet.