Let A and B be two square matrices of order 3 such that | A | = 3 and | B | = 2 . Then | A T A ( adj ( 2 A ) ) - 1 ( adj ( 4 B ) ) ( adj ( A B ) ) - 1 A A T | is equal to : [2024]

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Solution

Q.
Let $A$ and $B$ be two square matrices of order 3 such that $| A | = 3$ and $| B | = 2 .$ Then $| A^{T} A (adj {(2 A))}^{- 1} (adj (4 B)) (adj {(A B))}^{- 1} A A^{T} |$ is equal to : [2024]

1 108

2 32

3 64

4 81

Ans.
(3)

We have $| A^{T} A (adj {(2 A))}^{- 1} (adj (4 B)) (adj {(A B))}^{- 1} A A^{T} |$

$=$ $| A^{T} | | A | \frac{1}{| adj 2 A |} | adj 4 B | \frac{1}{| adj A B |} | A | | A^{T} |$

$= | A |^{2} \frac{1}{{(2^{2})}^{3} | adj A |} {(4^{2})}^{3} | adj B | \frac{1}{| adjA B |} | A |^{2}$ $[∵ | A | = | A^{T} | and adj kA = k^{n - 1} adj A, where n is order of A]$

$= | A |^{2} \frac{1}{2^{6} | A |^{2}} 4^{6} | B |^{2} \frac{1}{| A |^{2} | B |^{2}} | A |^{2}$ $[∵ | adj A | = | A |^{n - 1}, where n is order of A]$

$= \frac{4^{6}}{2^{6}} = 64$

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