FIND THE INVERSE OF A SQUARE MATRIX

Example 1 :

Solution :

So, AB  =  BA  =  I (Identity matrix)

Hence proved.

Example 2 :

Solution :

Since |A|  =  0, it is singular matrix, A^-1 does not exists.

So, there is no inverse matrix of A.

Hence proved.

Example 3 :

Solution :

Example 4 :

Solution :

Example 5 :

Solution :

Example 6 :

Solution :

|A| =  +2(0 - 4) - 3(-1 + 0) - 1(-1 + 0)

=  -8 + 3 + 1

=  -8 + 4

|A| =  -4

-4 ≠ 0

Since |A|  =  -4 ≠ 0, it is non singular matrix. A^-1 exists.

Example 7 :

Solution :

General term : 

b_ij  =  = |i – j|

 (where 1 ≤ i ≤ 3, 1 ≤ j ≤ 3)

Number of rows of the required matrix is 3.

Number of columns of the required matrix is 3.

Finding the elements :

Hence the required matrix is

Finding the inverse matrix of B :

|B| =  + 0(0 - 1) - 1(0 - 2) + 2(1 - 0)

=  2 + 2

=  4

Since |B|  =  4 ≠ 0, it is non singular matrix. B^-1 exists.

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