Question

$\left[\begin{array}{cc}1& 6\\ 7& 2\end{array}\right]$ = P + Q, where P is a symmetric & Q is a skew-symmetric, then P =

• A. $\left[\begin{array}{cc}1& \frac{13}{2}\\ \frac{13}{2}& 2\end{array}\right]$
• B. $\left[\begin{array}{cc}1& -\frac{13}{2}\\ -\frac{13}{2}& 2\end{array}\right]$
• C. $\left[\begin{array}{cc}1& -\frac{1}{2}\\ -\frac{1}{2}& 2\end{array}\right]$
• D. $\left[\begin{array}{cc}0& 13/2\\ 13/2& 0\end{array}\right]$

Answer 1

The correct option is A $\left[\begin{array}{cc}1& \frac{13}{2}\\ \frac{13}{2}& 2\end{array}\right]$

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 P i.e. the symmetric term is given by

$P=\frac{1}{2}(S+S^T)$

$P=\frac{1}{2}(\left[\begin{array}{cc}1& 6\\ 7& 2\end{array}\right]+\left[\begin{array}{cc}1& 7\\ 6& 2\end{array}\right])$

$P=\frac{1}{2}\left[\begin{array}{cc}2& 13\\ 13& 4\end{array}\right]$

$P=\left[\begin{array}{cc}1& \frac{13}{2}\\ \frac{13}{2}& 2\end{array}\right]$

hence the answer is option A