Question

If $A=\left(\begin{array}{cc}3& 5\\ 7& 9\end{array}\right)$ is written at $P+Q$, where $P$ is symmetric matrix and $Q$ is skew-symmetric matrix, then write the matrix $P$.

Answer 1

$A=\left[\begin{array}{cc}3& 5\\ 7& 9\end{array}\right]$

P is a symmetric matrix

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

Q is a skew symmetric matrix

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

$A^T=\left[\begin{array}{cc}3& 7\\ 5& 9\end{array}\right]$

$P=\frac{1}{2}\left[\begin{array}{cc}6& 12\\ 12& 18\end{array}\right]=\left[\begin{array}{cc}3& 6\\ 6& 9\end{array}\right]$.