If P and Q are symmetric matrices , then QP-PQ is
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → Skew-symmetric matrix
Since \( P^T=P \) and \( Q^T=Q \), the reversal law \( (XY)^T=Y^TX^T \) gives
\( (QP-PQ)^T=P^TQ^T-Q^TP^T \)
\( =PQ-QP \)
\( =-(QP-PQ) \)
So QP − PQ is skew-symmetric.
Why the other options are incorrect
Option (2) Singular matrix — Incorrect. Not in general: for \( P=\begin{bmatrix}1 & 0\\0 & 0\end{bmatrix} \) and \( Q=\begin{bmatrix}0 & 1\\1 & 0\end{bmatrix} \), \( QP-PQ=\begin{bmatrix}0 & -1\\1 & 0\end{bmatrix} \), whose determinant is 1.
Option (3) Zero matrix — Incorrect. QP − PQ is zero only when P and Q commute, which is not true in general.
Option (4) Symmetric matrix — Incorrect. It equals minus its transpose, so it is symmetric only when it is the zero matrix.
NCERT (2026-27 edition):
Symmetric and Skew Symmetric Matrices
