Question

If $\mathrm{A}=\left(\begin{array}{cc}3& 5\\ 7& 9\end{array}\right)$ 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. So, $P=\frac{1}{2}\left(A+A^T\right)$

Q is skew symmetric matrix. So, $Q=\frac{1}{2}\left(A-A^T\right)$

$$\begin{gathered} A^T=\left[\begin{array}{cc}3& 7\\ 5& 9\end{array}\right] \\ P=\frac{1}{2}\left[\begin{array}{cc}6& 12\\ 12& 18\end{array}\right]=\left[\begin{array}{cc}3& 6\\ 6& 9\end{array}\right] \end{gathered}$$