Question:
If $\mathrm{A}=\left(\begin{array}{ll}3 & 5 \\ 7 & 9\end{array}\right)$ is written as $A=P+Q$, where as $A=P+Q$, where $P$ is a symmetric matrix and $Q$ is skew symmetric matrix, then write the matrix $P$.
Solution:
$A=\left[\begin{array}{ll}3 & 5 \\ 7 & 9\end{array}\right]$
$P$ is symmetric matrix. So, $P=\frac{1}{2}\left(A+A^{T}\right)$
$Q$ is skew symmetric matrix. So, $Q=\frac{1}{2}\left(A-A^{T}\right)$
$A^{T}=\left[\begin{array}{ll}3 & 7 \\ 5 & 9\end{array}\right]$
$P=\frac{1}{2}\left[\begin{array}{cc}6 & 12 \\ 12 & 18\end{array}\right]=\left[\begin{array}{ll}3 & 6 \\ 6 & 9\end{array}\right]$
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