Let A = [ begin array ll 1 & 2\\ 1 & α end array ] and B = [ begin array ll 3 & 3\\ β & 2 end array ] . If A 2 - 4 A + I = O and B 2 - 5 B - 6 I = O then among the two statements : (S1) : [( B - A )( B + A ) ] T = [ begin array ll 13 & 15\\ 7 & 10 end array ] and (S2) : det (adj(A + B)) = -5,

Mathematics · Matrices

JEE Main 2026 — 2 April, Morning Shift — Question 23

Let $\mathbf{A} = \left[ \begin{array}{ll}1 & 2\\ 1 & \alpha \end{array} \right]$ and $\mathbf{B} = \left[ \begin{array}{ll}3 & 3\\ \beta & 2 \end{array} \right]$ . If $\mathbf{A}^2 - 4\mathbf{A} + \mathbf{I} = \mathbf{O}$ and $\mathbf{B}^2 - 5\mathbf{B} - 6\mathbf{I} = \mathbf{O}$ then among the two statements :

(S1) : $\left[(\mathbf{B} - \mathbf{A})(\mathbf{B} + \mathbf{A})\right]^{\mathrm{T}} = \left[ \begin{array}{ll}13 & 15\\ 7 & 10 \end{array} \right]$ and

(S2) : det (adj(A + B)) = -5,

Option A:
only (S1) is correct
Option B:
only (S2) is correct
Correct
Option C:
both (S1) and (S2) are correct
Option D:
both (S1) and (S2) are wrong

Answer: B

Step-by-step solution

$\quad A=\left[\begin{array}{ll}1 & 2\\ 1 & \alpha\end{array}\right]$ $A^{2}-4 A+I=0$ trace of matrix $\alpha+1=4$ $\alpha=3$ $B=\left[\begin{array}{ll}3 & 3\\ \beta & 2\end{array}\right]$ $\mathrm{B}^{2}-5 \mathrm{~B}-6 \mathrm{I}=0$ $\operatorname{det} .(B)=6-3 \beta=-6$ $3 \beta=12$ $\beta=4$ $\mathrm{B}-\mathrm{A}=\left[\begin{array}{cc}2 & 1\\ 3 & -1\end{array}\right] ; \mathrm{B}+\mathrm{A}=\left[\begin{array}{ll}4 & 5\\ 5 & 5\end{array}\right]$ $(B-A)(B+A)=\left[\begin{array}{cc}13 & 15\\ 7 & 10\end{array}\right]$ $\operatorname{adj}(\mathrm{A})=\left[\begin{array}{cc}3 & -2\\ -1 & 1\end{array}\right], \operatorname{adjB}=\left[\begin{array}{cc}2 & -3\\ -4 & 3\end{array}\right]$ $\operatorname{adj}(\mathrm{A}+\mathrm{B})=\left[\begin{array}{cc}5 & -5\\ -5 & 4\end{array}\right]$

$det (adj(A + B)) = -5,$

Answer key and solution verified before publishing.

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Exam: JEE Main 2026
Paper: 2 April, Morning Shift
Subject: Mathematics
Chapter: Matrices
Topic: Inverse of a Matrix