If \( P=\begin{bmatrix}1 & 2\\-1 & 3\end{bmatrix} \), \( Q=\begin{bmatrix}4 & 0\\1 & 5\end{bmatrix} \) and \( R=\begin{bmatrix}2 & 0\\1 & -2\end{bmatrix} \), then which of the following are TRUE?
(A) \( (P^T)^T=P^T \)
(B) \( (P-Q)R=PR-QR \)
(C) \( (P-Q)^T=Q^T-P^T \)
(D) \( (PQ)^T=Q^TP^T \)
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → (B) and (D) only
Transposing twice gives back the matrix itself, \( (P^T)^T=P \), and here P is not symmetric:
\( \displaystyle P=\begin{bmatrix}1 & 2\\-1 & 3\end{bmatrix} \)
\( \displaystyle P^T=\begin{bmatrix}1 & -1\\2 & 3\end{bmatrix}\ne P \)
The transpose of a difference is \( (P-Q)^T=P^T-Q^T \), while \( Q^T-P^T=-(P-Q)^T \). These are equal only if \( P-Q=O \), and
\( \displaystyle P-Q=\begin{bmatrix}-3 & 2\\-2 & -2\end{bmatrix}\ne O \)
Matrix multiplication distributes over subtraction, and the transpose of a product reverses the order (reversal law).
So \( (P^T)^T=P\ne P^T \), \( (P-Q)^T=P^T-Q^T\ne Q^T-P^T \), \( (P-Q)R=PR-QR \) and \( (PQ)^T=Q^TP^T \).
Checking the statements
(A) \( (P^T)^T=P^T \) — Incorrect. \( (P^T)^T=P \), and \( P\ne P^T \) since 2 ≠ −1.
(B) \( (P-Q)R=PR-QR \) — Correct. Distributive law, with R kept on the right.
(C) \( (P-Q)^T=Q^T-P^T \) — Incorrect. \( (P-Q)^T=P^T-Q^T \), and \( P-Q\ne O \).
(D) \( (PQ)^T=Q^TP^T \) — Correct. Reversal law for the transpose of a product.