Let A be a 3 × 3 matrix and det(A) = 2. If n = det t i m e s , then the remainder when n is divided by 9 is equal to ________ . [2024]

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Solution

Q.
Let $A$ be a $3 \times 3$ matrix and det(A) = 2. If $n = \det \underset{2024 - t i m e s}{\underset{⏟}{(a d j (a d j (. . . (a d j A))))}},$ then the remainder when $n$ is divided by 9 is equal to ________ . [2024]

Ans.
(7)

We know that $| adj A |$ $= | A |^{n - 1} = | A |^{2}$

$\Rightarrow | adj adj A | = | adj A |^{2} = | A |^{2^{2}}$

$n = \underset{2024 t i m e s}{\underset{⏟}{| a d j a d j . . . a d j A |}} = | A |^{2^{2024}}$

Here, $2^{2024} = 4^{1012} = {(3 + 1)}^{1012} = 3 k + 1$

$\Rightarrow n = | A |^{3 k + 1} = 2^{3 k + 1} [∵ | A | = 2]$

$= 2 \cdot 8^{k} = 2 {(9 - 1)}^{k} = 2 (9 λ - 1) = 18 λ - 2$

$= 18 (λ - 1) + 16 = 18 (λ - 1) + 9 + 7$

Hence, the required remainder is 7.

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