If P and Q are symmetric matrices of the same order, then \( (\mathrm{PQ}^{\mathrm{T}}-\mathrm{QP}^{\mathrm{T}}) \) is :
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → Skew symmetric matrix
Since \( \mathrm{P}^{\mathrm{T}}=\mathrm{P} \) and \( \mathrm{Q}^{\mathrm{T}}=\mathrm{Q} \), let
\( \displaystyle \mathrm{M}=\mathrm{PQ}^{\mathrm{T}}-\mathrm{QP}^{\mathrm{T}}=\mathrm{PQ}-\mathrm{QP} \)
Taking the transpose, with \( (\mathrm{XY})^{\mathrm{T}}=\mathrm{Y}^{\mathrm{T}}\mathrm{X}^{\mathrm{T}} \):
\( \displaystyle \mathrm{M}^{\mathrm{T}}=\mathrm{Q}^{\mathrm{T}}\mathrm{P}^{\mathrm{T}}-\mathrm{P}^{\mathrm{T}}\mathrm{Q}^{\mathrm{T}} \)
\( \displaystyle =\mathrm{QP}-\mathrm{PQ} \)
\( \displaystyle =-\mathrm{M} \)
So M is a skew symmetric matrix.
Why the other options are incorrect
Option (1) Unit matrix — Incorrect. A skew symmetric matrix has every diagonal entry 0, so M can never be the unit matrix.
Option (2) Symmetric matrix — Incorrect. A matrix that is both symmetric and skew symmetric must be the null matrix, and M is not null in general.
Option (3) Null matrix — Incorrect. M is null only when PQ = QP. For example, \( \mathrm{P}=\begin{bmatrix}1 & 0\\0 & 0\end{bmatrix} \) and \( \mathrm{Q}=\begin{bmatrix}0 & 1\\1 & 0\end{bmatrix} \) give \( \mathrm{PQ}-\mathrm{QP}=\begin{bmatrix}0 & 1\\-1 & 0\end{bmatrix} \).
Tip: For symmetric P and Q: PQ + QP is always symmetric and PQ − QP is always skew symmetric.
NCERT Exemplar:
Matrices
