JEE Main 2026 · 2 Apr · Shift 1 · Mathematics Q4

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MathematicsSingle correct+4 / -1Question 4 of 75

Let $A=\begin{bmatrix}1&2\\1&\alpha\end{bmatrix}$ and $B=\begin{bmatrix}3&3\\\beta&2\end{bmatrix}$ . If $A^2-4A+I=O$ and $B^2-5B-6I=O$ , then among the two statements :

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

and

(S2): $\det\left(\operatorname{adj}(A+B)\right)=-5$ ,

only (S1) is correct

only (S2) is correct

both (S1) and (S2) are correct

both (S1) and (S2) are wrong

Answer and worked solution

Official answer: B

Cayley-Hamilton. For a 2 × 2 matrix, $M^2-(\operatorname{tr}M)M+(\det M)I=O$ .

$A$ : $\operatorname{tr}A=1+\alpha=4$ , so $\alpha=3$ (and $\det A=1$ ✓).
$B$ : $\operatorname{tr}B=5$ ✓ and $\det B=6-3\beta=-6$ , so $\beta=4$ .

(S1). $B-A=\begin{bmatrix}2&1\\3&-1\end{bmatrix}$ and $B+A=\begin{bmatrix}4&5\\5&5\end{bmatrix}$ , so $(B-A)(B+A)=\begin{bmatrix}13&15\\7&10\end{bmatrix}$ . Its transpose $\begin{bmatrix}13&7\\15&10\end{bmatrix}$ differs from the given matrix: false.

(S2). $\det(A+B)=20-25=-5$ and $\det(\operatorname{adj}M)=(\det M)^{n-1}=-5$ : true.

(B)

Topics:Adjoint and inverse Matrix algebra

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JEE Main 2026 · 2 Apr · Shift 1, Mathematics question 4