Question

If $S=\left[\begin{array}{cc}6& -8\\ 2& 10\end{array}\right]=P+Q$, where $P$ is a symmetric & $Q$ is a skew -symmetric matrix, then $Q$ is

• A. $\left[\begin{array}{cc}0& 5\\ -5& 0\end{array}\right]$
• B. $\left[\begin{array}{cc}0& -5\\ 5& 0\end{array}\right]$
• C. $\left[\begin{array}{cc}0& 8\\ -8& 0\end{array}\right]$
• D. $\left[\begin{array}{cc}0& 6\\ -6& 0\end{array}\right]$

Answer 1

The correct option is B $\left[\begin{array}{cc}0& -5\\ 5& 0\end{array}\right]$

Any square matrix can be expressed as sum of symmetric and skew-symmetric as
$A=\frac{1}{2}(A+A^T)+\frac{1}{2}(A-A^T)$
Where the first term is symmetric and second term os skew-symmetric
Here $S=P+Q$
to find $Q$ i.e. the skew-symmetric term is given by
$Q=\frac{1}{2}(S-S^T)Q=\frac{1}{2}(\left[\begin{array}{cc}6& -8\\ 2& 10\end{array}\right]-\left[\begin{array}{cc}6& 2\\ -8& 10\end{array}\right])Q=\frac{1}{2}\left[\begin{array}{cc}0& -10\\ 10& 0\end{array}\right]Q=\left[\begin{array}{cc}0& -5\\ 5& 0\end{array}\right]$