Question

Let $L=\left[\begin{array}{ccc}2& 3& 5\\ 4& 1& 2\\ 1& 2& 1\end{array}\right]=P+Q,$ where $P$ is a symmetric matrix & $Q$ is a skew-symmetric matrix, then $P$ is equal to

• A. $\left[\begin{array}{ccc}3& 5& 6\\ 5& 6& 4\\ 9& 4& 3\end{array}\right]$
• B. $\left[\begin{array}{ccc}2& \frac{7}{2}& 3\\ \frac{7}{2}& 1& 2\\ 3& 2& 1\end{array}\right]$
• C. $\left[\begin{array}{ccc}6& 5& 4\\ 3& 6& 3\\ 5& 2& 5\end{array}\right]$
• D. $\left[\begin{array}{ccc}1& \frac{7}{2}& 3\\ \frac{7}{2}& 2& 1\\ 3& 2& 1\end{array}\right]$

Answer 1

The correct option is B $\left[\begin{array}{ccc}2& \frac{7}{2}& 3\\ \frac{7}{2}& 1& 2\\ 3& 2& 1\end{array}\right]$

Given,$P+Q=\left[\begin{array}{ccc}2& 3& 5\\ 4& 1& 2\\ 1& 2& 1\end{array}\right]\cdots \left(i\right)$
$P$ is symmetric and $Q$ is skew symmetric matrix. So, by property of transpose,
$P=P^T$ and $Q^T=-Q$
$\therefore$Taking transpose of equation $\left(i\right)$, we have: $P^T-Q=\left[\begin{array}{ccc}2& 4& 1\\ 3& 1& 2\\ 5& 2& 1\end{array}\right]\cdots \left(ii\right)$
Add $\left(i\right)$ and $\left(ii\right)$, we get
$P+P^T=\left[\begin{array}{ccc}4& 7& 6\\ 7& 2& 4\\ 6& 4& 2\end{array}\right]\Rightarrow 2P=\left[\begin{array}{ccc}4& 7& 6\\ 7& 2& 4\\ 6& 4& 2\end{array}\right]$
$\Rightarrow P=\left[\begin{array}{ccc}2& \frac{7}{2}& 3\\ \frac{7}{2}& 1& 2\\ 3& 2& 1\end{array}\right]$