Let A = 6 , where A is a 3 × 3 matrix. If adj ( 3 adj ( A 2 · adj ( 2 A ) ) ) = 2 m · 3 n , m , n N , then m + n is equal to [2026]

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Solution

Q.
Let $| A |$ $= 6$ , where A is a $3 \times 3 matrix.$ If $| adj (3 adj (A^{2} \cdot adj (2 A))) |$ $= 2^{m} \cdot 3^{n}, m, n \in N,$ then $m + n$ is equal to _________ . [2026]

Ans.
(62)

$adj (2 A) = 2^{2} adj A ∵ adj (k A) = k^{n - 1} adj (A)$

   $= 4 adj A$

Now, $A^{2} (adj (2 A)) = 4 A (adj A)$

$= 4 A$ $| A |$ $I_{3}$

   $= 24 A$

Now,   $3 adj (A^{2} (adj (2 A))) = 3 adj (24 A)$

   $= 3 {(24)}^{2} adj A$

Now,   $| adj (3 adj (A^{2} (adj (2 A)))) |$

$=$ $| adj (3 . {(24)}^{2} adj A) |$

$=$ $| (3 . {{(24)}^{2})}^{2} adj (adj A) |$

$=$ $| 3^{6} \cdot 2^{12} adj (adj A) |$

$= {(3^{6} \cdot 2^{12})}^{3}$ $| adj (adj A) |$

$= 3^{18} \cdot 2^{36} \cdot | A |^{4}$

$= 3^{22} \cdot 2^{40}$

$∴$ $m + n = 62$

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