Maths Question 4 – JEE-MAIN 2026

$B-A=\left[\begin{array}{cc}3& 3\\ 4& 2\end{array}\right]-\left[\begin{array}{cc}1& 2\\ 1& 3\end{array}\right]=\left[\begin{array}{cc}2& 1\\ 3& -1\end{array}\right]$. $B+A=\left[\begin{array}{cc}3& 3\\ 4& 2\end{array}\right]+\left[\begin{array}{cc}1& 2\\ 1& 3\end{array}\right]=\left[\begin{array}{cc}4& 5\\ 5& 5\end{array}\right]$. Now, calculate $(B-A)(B+A)$: $(B-A)(B+A)=\left[\begin{array}{cc}2& 1\\ 3& -1\end{array}\right]\left[\begin{array}{cc}4& 5\\ 5& 5\end{array}\right]=\left[\begin{array}{cc}(2)(4)+(1)(5)& (2)(5)+(1)(5)\\ (3)(4)+(-1)(5)& (3)(5)+(-1)(5)\end{array}\right]=\left[\begin{array}{cc}13& 15\\ 7& 10\end{array}\right]$. Finally, calculate $[(B-A)(B+A){]}^T$: $[(B-A)(B+A){]}^T=\left[\begin{array}{cc}13& 7\\ 15& 10\end{array}\right]$. Since $\left[\begin{array}{cc}13& 7\\ 15& 10\end{array}\right]\ne \left[\begin{array}{cc}13& 15\\ 7& 10\end{array}\right]$, statement (S1) is incorrect.

$A+B=\left[\begin{array}{cc}1& 2\\ 1& 3\end{array}\right]+\left[\begin{array}{cc}3& 3\\ 4& 2\end{array}\right]=\left[\begin{array}{cc}4& 5\\ 5& 5\end{array}\right]$. Next, calculate $det(A+B)$: $det(A+B)=(4)(5)-(5)(5)=20-25=-5$. For a $2\times 2$ matrix $M$, the determinant of its adjoint is given by $det(\text{adj}(M))=(det(M){)}^{2-1}=det(M)$. Therefore, $det(\text{adj}(A+B))=det(A+B)=-5$. Since the calculated value is $-5$, statement (S2) is correct. Based on the evaluation, (S1) is incorrect and (S2) is correct. Thus, only (S2) is correct.