If A = , verify that A (adj A) = (adj A) A = | A | I3 .
Solution
Example 1.2
If A = is non-singular, find A−1
Solution
We first find adj A. By definition, we get
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 ⇒ adj A = (adj A)T ⇒ adj A is symmetric
Verify the property ( AT )−1 = ( A−1 )T with A = .
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) −1 = B-1 A-1 is verified.
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
Prove that is orthogonal.
Similarly, we get ATA = I2 . Hence AAT = ATA = I2 ⇒ A is orthogonal.
If A = , is orthogonal, find a, b and c , and hence A−1.
If A is orthogonal, then AAT = AT A = I3 . So, we have
Related Topics
Privacy Policy, Terms and Conditions, DMCA Policy and Compliant
Copyright © 2018-2023 BrainKart.com; All Rights Reserved. Developed by Therithal info, Chennai.