Question

$L=\left[\begin{array}{ccc}2& 3& 5\\ 4& 1& 2\\ 1& 2& 1\end{array}\right]=P+Q$, $P$ 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. $\left[\begin{array}{ccc}2& 3.5& 3\\ 3.5& 1& 2\\ 3& 2& 1\end{array}\right]$
• C. $\left[\begin{array}{ccc}6& 5& 4\\ 3& 6& 3\\ 5& 2& 5\end{array}\right]$
• D. $\left[\begin{array}{ccc}6& 5& 4\\ 4& 5& 3\\ 3& 4& 3\end{array}\right]$

Answer 1

The correct option is B $\left[\begin{array}{ccc}2& 3.5& 3\\ 3.5& 1& 2\\ 3& 2& 1\end{array}\right]$

Given ,$P+Q=\left[\begin{array}{ccc}2& 3& 5\\ 4& 1& 2\\ 1& 2& 1\end{array}\right]$-----(i)

ans $P$ is symmetric and $Q$ is skew symmetric matrix. So, by property of transpose (7),

$P=P^T$ and $Q^T=-Q$

$\therefore$ equation (i), $P^T-Q=\left[\begin{array}{ccc}2& 4& 1\\ 3& 1& 2\\ 5& 2& 1\end{array}\right]$---------(ii)

Add (i) and (ii), we get

$P+P^T=\left[\begin{array}{ccc}4& 7& 6\\ 7& 2& 4\\ 6& 4& 2\end{array}\right],2P=\left[\begin{array}{ccc}4& 7& 6\\ 7& 2& 4\\ 6& 4& 2\end{array}\right]$

$\Rightarrow P=\left[\begin{array}{ccc}2& 3.5& 3\\ 3.5& 1& 2\\ 3& 2& 1\end{array}\right]$