Let A be a 3 × 3 matrix such that | adj ( adj ( adj A ) ) | = 12 4 . Then | A - 1 adj A | is equal to [2023]

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Solution

Q.
$Let A be a 3 \times 3 matrix such that$ $| adj (adj (adj A)) |$ $= 12^{4} . Then | A^{- 1} adj A | is equal to$ [2023]

1 1

2 $\sqrt{6}$

3 12

4 $2 \sqrt{3}$

Ans.
(4)

Given, $| adj (adj (adj A)) |$ $= 12^{4}$

$\Rightarrow$ $| A |^{{(n - 1)}^{3}}$ $= 12^{4}$

At $n = 3,$ $| A |^{8} = 12^{4} \Rightarrow | A |^{2} = 12 \Rightarrow$ $| A |$ $= 2 \sqrt{3}$

$∴$ $| A^{- 1} adj A |$ $=$ $| A^{- 1} |$ $\cdot$ $| adj A |$ $= \frac{1}{| A |} \cdot | A |^{2} =$ $| A |$ $= 2 \sqrt{3}$

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