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

Solution:

Given that $\mathrm{A}=\left[\begin{array}{rr}3 & -4 \\ 1 & -1\end{array}\right]$ ,to express as the sum of symmetric matrix $\mathrm{P}$ and skew symmetric matrix Q. $A=P+Q$
Where $\mathrm{P}=\frac{1}{2}\left(\mathrm{~A}+\mathrm{A}^{\prime}\right)$ and $\mathrm{Q}=\frac{1}{2}\left(\mathrm{~A}-\mathrm{A}^{\prime}\right)$ , we will find transpose of matrix $\mathrm{A}$
$\mathrm{A}^{\prime}=\left[\begin{array}{cc} 3 & 1 \\ -4 & -1 \end{array}\right]$
Using the above formulas
$\begin{array}{l} P=\frac{1}{2}\left(\mathrm{~A}+\mathrm{A}^{\prime}\right) \\ \Rightarrow \frac{1}{2}\left(\left[\begin{array}{rr} 3 & -4 \\ 1 & -1 \end{array}\right]+\left[\begin{array}{cc} 3 & 1 \\ -4 & -1 \end{array}\right]\right) \\ \Rightarrow \frac{1}{2}\left[\begin{array}{cc} 6 & -3 \\ -3 & -2 \end{array}\right] \end{array}$
$\begin{array}{l} \Rightarrow\left[\begin{array}{cc} 3 & \frac{-3}{2} \\ \frac{-3}{2} & -1 \end{array}\right] \\ Q=\frac{1}{2}\left(A-A^{\prime}\right) \\ \Rightarrow \frac{1}{2}\left(\left[\begin{array}{ll} 3 & -4 \\ 1 & -1 \end{array}\right]-\left[\begin{array}{cc} 3 & 1 \\ -4 & -1 \end{array}\right]\right) \\ \left.\Rightarrow \frac{1}{2} \mid \begin{array}{cc} 0 & -5 \\ 5 & 0 \end{array}\right] \end{array}$
$\begin{array}{l} \Rightarrow \frac{1}{2}\left[\begin{array}{cc} 0 & -5 \\ 5 & 0 \end{array}\right] \\ \Rightarrow\left[\begin{array}{cc} 0 & \frac{-5}{2} \\ \frac{5}{2} & 0 \end{array}\right] \end{array}$
Hence $A=P+Q$
$\Rightarrow\left[\begin{array}{cc} 3 & \frac{-3}{2} \\ \frac{-3}{2} & -1 \end{array}\right]+\left[\begin{array}{cc} 0 & \frac{-5}{2} \\ \frac{5}{2} & 0 \end{array}\right][\text { Matrix } A \text { as the sum of } P \text { and } \mathrm{Q}]$
$\begin{array}{l} \Rightarrow\left[\begin{array}{cc} 3 & \frac{-8}{2} \\ \frac{2}{2} & -1 \end{array}\right] \\ \Rightarrow\left[\begin{array}{cc} 3 & -4 \\ 1 & -1 \end{array}\right] \end{array}$