FINDING INVERSE OF 3X3 MATRIX EXAMPLES

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

AB  =   BA  =  In

then the matrix B is called an inverse of A.

Note :

Let A be square matrix of order n. Then, A−1 exists if and only if A is non-singular.

Finding Inverse of 2 x 2 Matrix

Example 1 :

Find the inverse (if it exists) of the following:

Since |A|  =  2  0, it is non singular matrix. A-1 exists.

Finding the Inverse of a 3x3 Matrix Examples

Example 2 :

Solution :

In order to find inverse of a matrix, first we have to find |A|.

|A|  =  5(25 - 1) - 1(5 - 1) + 1(1 - 5)

=  5(24 ) - 1(4) + 1(-4)

=  120 - 4 - 4

=  112

Since |A|  =  112  0, it is non singular matrix. A-1 exists.

Example 3 :

Solution :

In order to find inverse of a matrix, first we have to find |A|.

|A|  =  2(8 - 7) - 3(6 - 3) + 1(21 - 12)

=  2(1) - 3(3) + 1(9)

=  2 - 9 + 9

=  2

Since |A|  =  2  0, it is non singular matrix. A-1 exists.

Example 4 :

Solution :

Let A  =  F(α)

|A|  =  cos α [cos α - 0] - 0[0 - 0] + sin α[0 + sin α]

=  cos2α + sin2α

|A|  =  1

Hence proved.

Example 5 :

Solution :

=  A2 - 3A - 7I2

Finding the value of A2 :

Finding the value of 3A :

Finding the value of 7I2 :

A2 - 3A - 7I

Hence proved.

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