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=

• 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

Answer: (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 is skew-symmetric

Here $S=P+Q$

to find Q i.e. the skew-symmetric matrix is given by

$Q=\frac{1}{2}(S-S^T)$

$Q=\frac{1}{2}(\left[\begin{array}{cc}6& -8\\ 2& 10\end{array}\right]-\left[\begin{array}{cc}6& 2\\ -8& 10\end{array}\right])$

$Q=\frac{1}{2}\left[\begin{array}{cc}0& -10\\ 10& 0\end{array}\right]$

$Q=\left[\begin{array}{cc}0& -5\\ 5& 0\end{array}\right]$

Hence the answer is option B.