If A and B are symmetric matrices of the same order, then
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → (AB+BA) is a symmetric matrix.
Since A and B are symmetric, \(A^T = A\) and \(B^T = B\). By the reversal law \((AB)^T = B^TA^T\):
\((AB + BA)^T = (AB)^T + (BA)^T\)
\(= B^TA^T + A^TB^T\)
\(= BA + AB\)
So AB + BA equals its own transpose: it is symmetric.
Why the other options are incorrect
Option (1) (AB+BA) is skew symmetric matrix. — Incorrect. A matrix that is both symmetric and skew-symmetric must be the zero matrix, and AB + BA is not zero in general (A = B = I gives 2I).
Option (2) (AB+BA) is always a diagonal matrix. — Incorrect. With B = I, AB + BA = 2A, which is not diagonal when A has non-zero entries off the diagonal.
Option (4) (AB-BA) is a symmetric matrix. — Incorrect. \((AB - BA)^T = BA - AB = -(AB - BA)\), so AB − BA is skew-symmetric.