JEE 2025 — Mathematics PYQ

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Question

JEE 2025 — Mathematics PYQ

JEE | Mathematics | 2025

Let $a \in R$ and $A$ be a matrix of order $3 \times 3$ such that $det (A) = - 4$ and $A + I = 12 a am p; a am p; 1 am p; 1 am p; 1 am p; 0 am p; 2$ , where $I$ is the identity matrix of order $3 \times 3$ . If $det ((a + 1) adj ((a - 1) A))$ is $2^{m} 3^{n}, m, n \in {0, 1, 2, \dots, 20}$ , then $m + n$ is equal to:

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Explanation

$A = 12 a am p; a am p; 1 am p; 1 am p; 1 am p; 0 am p; 2 - I = 02 a am p; a am p; 0 am p; 1 am p; 1 am p; 0 am p; 1$
$∣ A ∣ = 02 a am p; a am p; 0 am p; 1 am p; 1 am p; 0 am p; 1 = 2 - 2 a$
Given, $∣ A ∣ = - 4 \Rightarrow 2 - 2 a = - 4 \Rightarrow a = 3$
$∣ (a + 1) adj ((a - 1) A) ∣ = ∣4 adj 3 A ∣$

$A = 02 a am p; a am p; 0 am p; 1 am p; 1 am p; 0 am p; 1$
$∣ A ∣ = 2 - 2 a = - 4 \Rightarrow a = 3$
$∣ (a + 1) adj ((a - 1) A) ∣ = ∣4 adj 3 A ∣$
$= 4^{3} ∣ adj 3 A ∣ = 4^{3} \times ∣3 A ∣^{3 - 1} = 64∣3 A ∣^{2}$
$= 64 \times (3^{3})^{2} ∣ A ∣^{2} = 2^{6} \times 3^{6} \times 16$
$2^{m} \times 3^{n} = 2^{10} \times 3^{6} \Rightarrow m = 10, n = 6$
$m + n = 16$

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