The Inverse of a Square Matrix

Tutorial on how to find the inverse of a square matrix.

Definition of The Matrix Identity

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.


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Updated: 2 April 2013

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