If A and B are two symmetric matrices of the same order, then
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → AB – BA is a skew symmetric matrix
Since A and B are symmetric, \( A^T=A \) and \( B^T=B \). By the reversal law \( (AB)^T=B^TA^T \):
\( \displaystyle (AB-BA)^T=B^TA^T-A^TB^T \)
\( \displaystyle =BA-AB \)
\( \displaystyle =-(AB-BA) \)
So AB − BA is skew symmetric.
Why the other options are incorrect
Option (1) AB + BA is a skew symmetric matrix — Incorrect. \( (AB+BA)^T=BA+AB \), so AB + BA is symmetric, not skew symmetric.
Option (2) AB – BA is a symmetric matrix — Incorrect. The transpose of AB − BA is BA − AB, its negative, so it is skew symmetric; it is symmetric only in the special case AB = BA, when it is the zero matrix.
Option (4) A + B is a skew symmetric matrix — Incorrect. \( (A+B)^T=A+B \), so A + B is symmetric.
NCERT (2026-27 edition):
Symmetric and Skew Symmetric Matrices
