Express $\left[ \begin{matrix}

3 & \phantom{-}5 \\

1 & -1 \\

\end{matrix} \right]$ as the sum of a symmetric and skew symmetric matrices.

Using the formula

A = $\dfrac{1}{2} [A + A']$ + $\dfrac{1}{2} [A - A']$


We have

A = $\left[ \begin{matrix}

3 & \phantom{-}5 \\

1 & -1 \\

\end{matrix} \right]$ and A' = $\left[ \begin{matrix}


3 & \phantom{-}1 \\

5 & -1 \\

\end{matrix} \right]$


Symmetric Matrix

$\dfrac{1}{2} [A + A']$ =
$\dfrac{1}{2}$
$\left[ \begin{matrix}

3 & \phantom{-}5 \\

1 & -1 \\

\end{matrix} \right]$ +
$\left[ \begin{matrix}


3 & \phantom{-}1 \\

5 & -1 \\

\end{matrix} \right]$ = $\dfrac{1}{2}$ $\left[ \begin{matrix}


6 & \phantom{-}6 \\

6 & -2 \\

\end{matrix} \right]$ = $\left[ \begin{matrix}


3 & \phantom{-}3 \\

3 & -1 \\

\end{matrix} \right]$


Skew symmetric Matrix

$\dfrac{1}{2} [A - A']$ =
$\dfrac{1}{2}$
$\left[ \begin{matrix}

3 & \phantom{-}5 \\

1 & -1 \\

\end{matrix} \right]$ -
$\left[ \begin{matrix}


3 & \phantom{-}1 \\

5 & -1 \\

\end{matrix} \right]$ = $\dfrac{1}{2}$ $\left[ \begin{matrix}


0 & \phantom{-}4 \\

-4 & \phantom{-}0 \\

\end{matrix} \right]$ = $\left[ \begin{matrix}


0 & \phantom{-}2 \\

-2 & \phantom{-}0 \\

\end{matrix} \right]$


A = sum of symmetric matrix and skew symmetric matrix

$\left[ \begin{matrix}


3 & \phantom{-}3 \\

3 & -1 \\

\end{matrix} \right]$ + $\left[ \begin{matrix}


0 & \phantom{-}2 \\

-2 & \phantom{-}0 \\

\end{matrix} \right]$ = $\left[ \begin{matrix}

3 & \phantom{-}5 \\

1 & -1 \\

\end{matrix} \right]$