Question

If $A=\left(\begin{array}{cc}3& 5\\ 7& 9\end{array}\right)$ is written as $A=P+Q$,
where $P$ is a 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 symmetric matrix.

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

$Q$ is skew symmetric matrix.

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

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

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

$\Rightarrow$ $P=\frac{1}{2}(\left[\begin{array}{cc}3& 5\\ 7& 9\end{array}\right]+\left[\begin{array}{cc}3& 7\\ 5& 9\end{array}\right])$

$\Rightarrow$ $P=\frac{1}{2}\left[\begin{array}{cc}3+3& 5+7\\ 7+5& 9+9\end{array}\right]$

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

$\Rightarrow$ $P=\left[\begin{array}{cc}3& 6\\ 6& 9\end{array}\right]$