If $\begin{bmatrix} 2 & 0 \\ 5 & 4 \end{bmatrix} = P + Q$, where $P$ is a symmetric and $Q$ is a skew symmetric matrix, then $Q$ is equal to:
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → $\begin{bmatrix} 0 & -\frac{5}{2} \\ \frac{5}{2} & 0 \end{bmatrix}$ ##
$\begin{bmatrix} 2 & 0 \\ 5 & 4 \end{bmatrix} = P + Q = \frac{1}{2}(A + A^T) + \frac{1}{2}(A - A^T)$
$2A = \begin{bmatrix} 2 & 0 \\ 5 & 4 \end{bmatrix}$
$2A^T = \begin{bmatrix} 2 & 5 \\ 0 & 4 \end{bmatrix}$
$Q = \frac{A - A^T}{2}$
$= \frac{1}{2} \left( \begin{bmatrix} 2 & 0 \\ 5 & 4 \end{bmatrix} - \begin{bmatrix} 2 & 5 \\ 0 & 4 \end{bmatrix} \right)$
$= \frac{1}{2} \begin{bmatrix} 0 & -5 \\ 5 & 0 \end{bmatrix} = \begin{bmatrix} 0 & -\frac{5}{2} \\ \frac{5}{2} & 0 \end{bmatrix}$