Here

$B^{\prime}=\left[\begin{array}{rrr}2 & -1 & 1 \\ -2 & 3 & -2 \\ -4 & 4 & -3\end{array}\right]$

Let      $P=\frac{1}{2}\left(B+B^{\prime}\right)=\frac{1}{2}\left[\begin{array}{ccc}4 & -3 & -3 \\ -3 & 6 & 2 \\ -3 & 2 & -6\end{array}\right]$  $=\left[\begin{array}{ccc}2 & \frac{-3}{2} & \frac{-3}{2} \\ \frac{-3}{2} & 3 & 1 \\ \frac{-3}{2} & 1 & -3\end{array}\right]$

Now      $P^{\prime}=\left[\begin{array}{ccc}2 & \frac{-3}{2} & \frac{-3}{2} \\ \frac{-3}{2} & 3 & 1 \\ \frac{-3}{2} & 1 & -3\end{array}\right]=P$

Thus     $P=\frac{1}{2}\left(B+B^{\prime}\right)$ is a symmetric matrix.

Also, let   $\mathrm{Q}=\frac{1}{2}\left(\mathrm{B}-\mathrm{B}^{\prime}\right)=\frac{1}{2}\left[\begin{array}{rrr}0 & -1 & -5 \\ 1 & 0 & 6 \\ 5 & -6 & 0\end{array}\right]$ $=\left[\begin{array}{ccc}0 & \frac{-1}{2} & \frac{-5}{2} \\ \frac{1}{2} & 0 & 3 \\ \frac{5}{2} & -3 & 0\end{array}\right]$

Then     $\mathrm{Q}^{\prime}=\left[\begin{array}{ccc}0 & \frac{1}{2} & \frac{5}{3} \\ \frac{-1}{2} & 0 & -3 \\ \frac{-5}{2} & 3 & 0\end{array}\right]=-\mathrm{Q}$

Thus   $Q=\frac{1}{2}\left(B-B^{\prime}\right)$ is a skew symmetric matrix.

Now     $P+Q=\left[\begin{array}{ccc}2 & \frac{-3}{2} & \frac{-3}{2} \\ \frac{-3}{2} & 3 & 1 \\ \frac{-3}{2} & 1 & -3\end{array}\right]$  $\left[\begin{array}{ccc}0 & \frac{-1}{2} & \frac{-5}{2} \\ \frac{1}{2} & 0 & 3 \\ \frac{5}{2} & -3 & 0\end{array}\right]=\left[\begin{array}{ccc}2 & -2 & -4 \\ -1 & 3 & 4 \\ 1 & -2 & -3\end{array}\right]=B$

Thus, $B$ is represented as the sum of a symmetric and a skew symmetric matrix.