If A and B are symmetric matrices then (AB + BA) is:

The correct answer is Option (1) → Symmetric matrix

A and B are symmetric, so \(A^T = A\) and \(B^T = B\). Using the reversal law \((AB)^T = B^TA^T\):

    \((AB + BA)^T = (AB)^T + (BA)^T\)
    \(= B^TA^T + A^TB^T\)
    \(= BA + AB\)
    \(= AB + BA\)

The matrix equals its own transpose, so AB + BA is symmetric.

It need not be diagonal or scalar. For example, take the symmetric matrices

    \( \displaystyle A = \begin{bmatrix}1 & 1\\ 1 & 1\end{bmatrix},\quad B = I \)

Then AB + BA = 2A, which is not diagonal.