If A and B are symmetric matrices of the same order, then which one of the following is true?
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → AB + BA is a symmetric matrix.
A and B are symmetric, so \(A' = A\) and \(B' = B\). Using the reversal law \((AB)' = B'A'\):
\((AB + BA)' = (AB)' + (BA)'\)
\(= B'A' + A'B'\)
\(= BA + AB\)
So \((AB + BA)' = AB + BA\), and AB + BA is symmetric.
Why the other options are incorrect
Option (1) A + B is a skew-symmetric matrix — Incorrect. \((A + B)' = A' + B' = A + B\), so A + B is symmetric, not skew-symmetric (unless it is the zero matrix).
Option (2) A − B is skew-symmetric matrix. — Incorrect. \((A - B)' = A' - B' = A - B\), so A − B is also symmetric.
Option (3) AB − BA is a symmetric matrix. — Incorrect. \((AB - BA)' = B'A' - A'B' = BA - AB = -(AB - BA)\), so AB − BA is skew-symmetric.