Let a in mathbb R and A be a matrix of order 3 × 3 such that det (A)=-4 and A+I= [begin array lll 1 & a & 1\\ 2 & 1 & 0\\ a & 1 & 2end array ] , where I is the identity matrix of order 3 × 3 . If det ((a+1) adj((a -1) A)) is 2 m 3 n , m, n in\ 0,1,2, ldots, 20\ , then m+n is equal to

Mathematics · Matrices

JEE Main 2025 — 2 April, Morning Shift — Question 23

Let $a \in \mathbb{R}$ and $A$ be a matrix of order $3 \times 3$ such that $\operatorname{det}(A)=-4$ and $A+I=\left[\begin{array}{lll}1 & a & 1\\ 2 & 1 & 0\\ a & 1 & 2\end{array}\right]$ , where $I$

is the identity matrix of order $3 \times 3$ . If $det ((a+1) adj((a -1) A))$ is $2^{m} 3^{n}, m, n \in\{0,1,2, \ldots, 20\}$ , then

$m+n$ is equal to

Option A:
$16$
Correct
Option B:
$17$
Option C:
$14$
Option D:
$15$

Answer: A

Step-by-step solution

$A+I=\left[\begin{array}{lll}1 & a & 1\\ 2 & 1 & 0\\ a & 1 & 2\end{array}\right]$

$A=\left[\begin{array}{lll}0 & a & 1\\ 2 & 0 & 0\\ a & 1 & 1\end{array}\right]$

$|A|=-4$

$-2 a+2=-4$

$\Rightarrow \quad a=3$

$|4 \operatorname{adj}(2 A)|=4^{3}|\operatorname{adj}(2 A)|$

$=4^{3}|2 A|^{2}$

$=4^{3} \times 2^{6}|A|^{2}$

$=4^{3} \times 2^{6 \times}(-4)^{2}$

$=2^{16} \times 3^{0}$

$\Rightarrow m+n=16$

Answer key and solution verified before publishing.

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Exam: JEE Main 2025
Paper: 2 April, Morning Shift
Subject: Mathematics
Chapter: Matrices
Topic: Adjoint of a Square Matrix

Let A= [beginarrayccalpha & -1\\ 6 & betaendarray ], alpha 0 , such that det(A)=0 and alpha+beta=1 . If I denotes 2 × 2 identity matrix…

Answer: A

Step-by-step solution

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