Inverse of a Non-Singular Square Matrix: Solved Example Problems

Example 1.2

If A =  is non-singular, find A−1

Solution

We first find adj A. By definition, we get 

Example 1.3

Find the inverse of the matrix 

Solution

Example 1.4

If A is a non-singular matrix of odd order, prove that |adj A| is positive.

Solution

Let A be a non-singular matrix of order 2m + 1 , where m = 0,1, 2, .. . . Then, we get |A| ≠ 0 and, by theorem 1.9 (ii), we have |adj A| = |A|(2m+1)-1 = |A|2m 

Since |A|2m is always positive, we get that |adj A| is positive.

Example 1.5

Find a matrix  A if adj( A) = 

Solution

Example 1.6

If adj A = find A−1.

Solution

Example1.7

If A is symmetric, prove that adj A is also symmetric.

Solution

Suppose A is symmetric. Then, AT = A and so, by theorem 1.9 (vi), we get

adj (AT) = (adj A) T  â‡’ adj A = (adj A)T â‡’ adj A is symmetric

Example 1.8

Verify the property  ( AT  )−1  = ( A−1 )T  with  A = .

Solution

For the given A, we get |A |= (2) (7) -  (9)(1) = 14 − 9 = 5 .

From (1) and (2), we get (A-1) = (AT)-1. Thus, we have verified the given property.

Example 1.9

Verify ( AB)−1 = B−1 A−1 with 

Solution

As the matrices in (1) and (2) are same, (AB) âˆ’1 = B-1 A-1 is verified.

Example 1.10

If  A  = , find x and y such that A2 + xA + yI2=O2, Hence, find A−1.

Solution

So, we get 22 + 4x + y =0, 31+5x+y=0, 27+3x=0 and 18+2x=0

Hence x = −9 and y =14.Then, we get A2 - 9A + 14I2 = O2

Post-multiplying this equation by A−1 , we get A – 9I2 + 14A-1 = O2.  Hence, we get

Example 1.11

Prove that  is orthogonal.

Solution

Similarly, we get ATA = I2 . Hence AAT = ATA = I2 â‡’ A is orthogonal.

Example 1.12

If A = , is orthogonal, find a, b and c , and hence A−1.

Solution

If A is orthogonal, then AAT = AT A = I3 . So, we have