Let a ∈ R and A be a matrix of order 3 × 3 such that det(A)=-4 and A+I=[ array lll 1 & a & 1 2 & 1 & 0 a & 1 & 2 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 ∈ \

Correct answer: D. 16 A=[ array lll 1 & a & 1 2 & 1 & 0 a & 1 & 2 array ]- I =[ array lll 0 & a & 1 2 & 0 & 0 a & 1 & 1 array ] aligned & |A|=-4 2-2 a=-4 a=3 & |(a+1) adj(a-1) A|=|4 adj 3 ~A| & =4^3|adj 3 ~A| & =4^3 ×|3 ~A|^ 3-1 =64|3 ~A|^2 & =64 ×(3^3)^2|~A|^2 & =2^6 × 3^6 × 16 & 2^ m × 3^ n =2^ 10 × 3^6 & ~m=10, n=6 & ~m+n=16 aligned

JEE Main 20252 AprilMorning Shift

Exam: JEE Main
Year: 2025
Date: 2 April
Shift: Morning Shift
Subject: Mathematics
Chapter: Matrices

JEE Main 2025 · 2 April · Morning Shift · Mathematics · Matrices

Let a ∈ R and A be a matrix of order 3 × 3 such that det(A)=-4 and...

Let

\mathrm{a} \in \mathbf{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

\operatorname{det}((a+1) \operatorname{adj}((a-1) A))

2^m 3^n, m, n \in

\{0,1,2, \ldots .20\}

, then

\mathrm{m}+\mathrm{n}

is equal to :

Options

Correct answer

D. 16

Step-by-step solution

\mathrm{A}=\left[

\begin{array}{lll}1 & \mathrm{a} & 1 \\ 2 & 1 & 0 \\ \mathrm{a} & 1 & 2\end{array}

\right]-

\mathrm{I}

=\left[

\begin{array}{lll}0 & \mathrm{a} & 1 \\ 2 & 0 & 0 \\ \mathrm{a} & 1 & 1\end{array}

\right]

\begin{aligned} & |\mathrm{A}|=-4 \Rightarrow 2-2 \mathrm{a}=-4 \Rightarrow \mathrm{a}=3 \\ & |(\mathrm{a}+1) \operatorname{adj}(\mathrm{a}-1) \mathrm{A}|=|4 \operatorname{adj} 3 \mathrm{~A}| \\ & =4^3|\operatorname{adj} 3 \mathrm{~A}| \\ & =4^3 \times|3 \mathrm{~A}|^{3-1}=64|3 \mathrm{~A}|^2 \\ & =64 \times\left(3^3\right)^2|\mathrm{~A}|^2 \\ & =2^6 \times 3^6 \times 16 \\ & 2^{\mathrm{m}} \times 3^{\mathrm{n}}=2^{10} \times 3^6 \\ & \therefore \mathrm{~m}=10, \mathrm{n}=6 \\ & \Rightarrow \mathrm{~m}+\mathrm{n}=16\end{aligned}

Practice Matrices on Quantrex Academy →

More from Matrices