JEE Main Mathematics Question (2026) — Solution

Step-by-step solution

For a 2 2 matrix M, the characteristic equation is given by M^2 - Tr (M)M + (M)I = O. For matrix A = bmatrix 1 & 2 \\ 1 & bmatrix , we are given A^2 - 4A + I = O. Comparing the trace, we get Tr (A) = 1 + = 4 = 3. Thus, A = bmatrix 1 & 2 \\ 1 & 3 bmatrix . For matrix B = bmatrix 3 & 3 \\ & 2 bmatrix , we are given B^2 - 5B - 6I = O. Comparing the determinant, we get (B) = 6 - 3 = -6 3 = 12 = 4. Thus, B = bmatrix 3 & 3 \\ 4 & 2 bmatrix . Evaluating Statement (S1): B - A = bmatrix 3 & 3 \\ 4 & 2 bmatrix - bmatrix 1 & 2 \\ 1 & 3 bmatrix = bmatrix 2 & 1 \\ 3 & -1 bmatrix B + A = bmatrix 3 & 3 \\ 4 & 2 bmatrix + bmatrix 1 & 2 \\ 1 & 3 bmatrix = bmatrix 4 & 5 \\ 5 & 5 bmatrix (B-A)(B+A) = bmatrix 2 & 1 \\ 3 & -1 bmatrix bmatrix 4 & 5 \\ 5 & 5 bmatrix = bmatrix 2(4)+1(5) & 2(5)+1(5) \\ 3(4)-1(5) & 3(5)-1(5) bmatrix = bmatrix 13 & 15 \\ 7 & 10 bmatrix [(B-A)(B+A)]^T = bmatrix 13 & 15 \\ 7 & 10 bmatrix ^T = bmatrix 13 & 7 \\ 15 & 10 bmatrix Since bmatrix 13 & 7 \\ 15 & 10 bmatrix bmatrix 13 & 15 \\ 7 & 10 bmatrix , Statement (S1) is incorrect. Evaluating Statement (S2): For any 2 2 matrix M, ( adj (M)) = (M)^ 2-1 = (M). ( adj (A+B)) = (A+B) (A+B) = bmatrix 4 & 5 \\ 5 & 5 bmatrix = 4(5) - 5(5) = 20 - 25 = -5 Thus, ( adj (A+B)) = -5. Statement (S2) is correct. Therefore, only (S2) is correct. Answer: only (S2) is correct