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 matrices.
This means:
$A' = A$ and $B' = B$.
Check each option:
1) $(A + B)' = A' + B' = A + B$ → symmetric, not skew-symmetric. ✘
2) $(A - B)' = A' - B' = A - B$ → symmetric, not skew-symmetric. ✘
3) $(AB - BA)' = B'A' - A'B' = BA - AB = -(AB - BA)$ → skew-symmetric, not symmetric. ✘
4) $(AB + BA)' = B'A' + A'B' = BA + AB = AB + BA$ → symmetric. ✔
Correct statement: AB + BA is a symmetric matrix.