Let A be an n n matrix such that | A | = 2. If the determinant of the matrix Adj ( 2 · Adj ( 2 A - 1 ) ) is 2 84 , then n is equal to ______. [2023]

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Solution

Q.
Let A be a $n \times n$ matrix such that $| A |$ = 2. If the determinant of the matrix $Adj (2 \cdot Adj (2 A^{- 1}))$ is $2^{84}$ , then $n$ is equal to ______. [2023]

Ans.
(5)

$| Adj (2 Adj (2 A^{- 1})) |$

$=$ $| 2 (Adj (2 A^{- 1})) |^{n - 1}$ $= 2^{n (n - 1)}$ $| Adj (2 A^{- 1}) |^{n - 1}$

$= 2^{n (n - 1)}$ $| (2 A^{- 1}) |^{(n - 1) (n - 1)}$ $= 2^{n (n - 1) + n (n - 1) (n - 1)} \cdot \frac{1}{| A |^{{(n - 1)}^{2}}}$

$= \frac{2^{n (n - 1) + n (n - 1) (n - 1)}}{2^{{(n - 1)}^{2}}} 2^{n (n - 1) + n {(n - 1)}^{2} - {(n - 1)}^{2}}$

$= 2^{(n - 1) (n^{2} - n + 1)} = 2^{84}$

$\Rightarrow (n - 1) (n^{2} - n + 1) = 84$

$\Rightarrow (n - 5) (n^{2} + 3 n + 17) = 0 \Rightarrow n = 5$

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