Consider the matrices \(P=\begin{bmatrix}0 & 3\\ -3 & 0\end{bmatrix}\), \(Q=\begin{bmatrix}2 & -5\\ -5 & 0\end{bmatrix}\) and \(R=\begin{bmatrix}0 & 1\\ 1 & 0\end{bmatrix}\).
(A) P is a skew-symmetric matrix
(B) Q is a symmetric matrix
(C) R is a diagonal matrix
(D) \(R^2 = I\), where I is the identity matrix of order 2
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → (A), (B) and (D) only
A square matrix M is symmetric if \(M' = M\) and skew-symmetric if \(M' = -M\). A diagonal matrix has every entry off the main diagonal equal to zero.
The transposes of P and Q:
\( \displaystyle P' = \begin{bmatrix}0 & -3\\ 3 & 0\end{bmatrix} = -P \)
\( \displaystyle Q' = \begin{bmatrix}2 & -5\\ -5 & 0\end{bmatrix} = Q \)
The square of R:
\( \displaystyle R^2 = \begin{bmatrix}0 & 1\\ 1 & 0\end{bmatrix}\begin{bmatrix}0 & 1\\ 1 & 0\end{bmatrix} \)
\( \displaystyle = \begin{bmatrix}0+1 & 0+0\\ 0+0 & 1+0\end{bmatrix} \)
\( \displaystyle = \begin{bmatrix}1 & 0\\ 0 & 1\end{bmatrix} = I \)
So \(P' = -P\), \(Q' = Q\), R has off-diagonal entries 1, and \(R^2 = I\).
Checking the statements
(A) P is a skew-symmetric matrix — Correct. \(P' = -P\).
(B) Q is a symmetric matrix — Correct. \(Q' = Q\).
(C) R is a diagonal matrix — Incorrect. Its off-diagonal entries are 1, not 0.
(D) \(R^2 = I\), where I is the identity matrix of order 2 — Correct. The product \(R \times R\) is \(\begin{bmatrix}1 & 0\\ 0 & 1\end{bmatrix}\).