Express the matrix $\mathrm{A}=\left[\begin{array}{ccc}-1 & 5 & 1 \\ 2 & 3 & 4 \\ 7 & 0 & 9\end{array}\right]$ as the sum of a symmetric and a skew-symmetric matrix.

Solution:

Given that $\mathrm{A}=\left[\begin{array}{ccc}-1 & 5 & 1 \\ 2 & 3 & 4 \\ 7 & 0 & 9\end{array}\right]$ , to express as sum of symmetric matrix $\mathrm{P}$ and skew symmetric matrix $\mathrm{Q}$
$A=P+Q$
Where $\mathrm{P}=\frac{1}{2}\left(\mathrm{~A}+\mathrm{A}^{\prime}\right)$ and $\mathrm{Q}=\frac{1}{2}\left(\mathrm{~A}-\mathrm{A}^{\prime}\right)$
First, we need to find $A^{\prime}$
$\mathrm{A}^{\prime}=\left[\begin{array}{ccc} -1 & 2 & 7 \\ 5 & 3 & 0 \\ 1 & 4 & 9 \end{array}\right]$
Using the above mentioned formulas
$\mathrm{P}=\frac{1}{2}\left(\mathrm{~A}+\mathrm{A}^{\prime}\right)$
$\Rightarrow \frac{1}{2}\left(\left[\begin{array}{ccc}-1 & 5 & 1 \\ 2 & 3 & 4 \\ 7 & 0 & 9\end{array}\right]+\left[\begin{array}{ccc}-1 & 2 & 7 \\ 5 & 3 & 0 \\ 1 & 4 & 9\end{array}\right]\right)$
$\Rightarrow \frac{1}{2}\left[\begin{array}{ccc}-2 & 7 & 8 \\ 7 & 6 & 4 \\ 8 & 4 & 18\end{array}\right]$
$\Rightarrow\left[\begin{array}{ccc}-1 & \frac{7}{2} & 4 \\ \frac{7}{2} & 3 & 2 \\ 4 & 2 & 9\end{array}\right]$
$Q=\frac{1}{2}\left(A-A^{\prime}\right)$
$\Rightarrow \frac{1}{2}\left(\left[\begin{array}{ccc}-1 & 5 & 1 \\ 2 & 3 & 4 \\ 7 & 0 & 9\end{array}\right]-\left[\begin{array}{ccc}-1 & 2 & 7 \\ 5 & 3 & 0 \\ 1 & 4 & 9\end{array}\right]\right)$
$\Rightarrow \frac{1}{2}\left[\begin{array}{ccc}0 & 3 & -6 \\ -3 & 0 & 4 \\ 6 & -4 & 0\end{array}\right]$
$\Rightarrow\left[\begin{array}{ccc}0 & \frac{3}{2} & -3 \\ \frac{-3}{2} & 0 & 2 \\ 3 & -2 & 0\end{array}\right]$
Now $A=P+Q$
$\Rightarrow\left[\begin{array}{ccc}-1 & \frac{7}{2} & 4 \\ \frac{7}{2} & 3 & 2 \\ 4 & 2 & 9\end{array}\right]+\left[\begin{array}{ccc}0 & \frac{3}{2} & -3 \\ \frac{-3}{2} & 0 & 2 \\ 3 & -2 & 0\end{array}\right]$ [Matrix A as sum of $\mathbf{P}$ and $\mathrm{Ql}$
$\Rightarrow\left[\begin{array}{ccc}-1 & \frac{10}{2} & 1 \\ \frac{4}{2} & 3 & 4 \\ 7 & 0 & 9\end{array}\right]$
$A=\left[\begin{array}{ccc} -1 & 5 & 1 \\ 2 & 3 & 4 \\ 7 & 0 & 9 \end{array}\right]$