Express the matrix $\left[ {\begin{array}{cccccccccccccccccccc}2&3&1\\1&{ - 1}&2\\4&1&2\end{array}} \right]$ as the sum of a symmetric and a skew-symmetric matrix.

We have, $A = \left[ {\begin{array}{cccccccccccccccccccc}2&3&1\\1&{ - 1}&2\\4&1&2\end{array}} \right]$
$\therefore$ ${A^\prime } = \left[ {\begin{array}{cccccccccccccccccccc}2&1&4\\3&{ - 1}&1\\1&2&2\end{array}} \right]$

Now, $\frac{{A + {A^\prime }}}{2} = \frac{1}{2}\left[ {\begin{array}{cccccccccccccccccccc}4&4&5\\4&{ - 2}&3\\5&3&4\end{array}} \right]$

$= \left[ {\begin{array}{cccccccccccccccccccc}2&2&{\frac{5}{2}}\\2&{ - 1}&{\frac{3}{2}}\\{\frac{5}{2}}&{\frac{3}{2}}&2\end{array}} \right]$

and $\frac{{A - {A^\prime }}}{2} = \frac{1}{2}\left[ {\begin{array}{cccccccccccccccccccc}0&2&{ - 3}\\{ - 2}&0&1\\3&{ - 1}&0\end{array}} \right] = \left[ {\begin{array}{cccccccccccccccccccc}0&1&{\frac{{ - 3}}{2}}\\{ - 1}&0&{\frac{1}{2}}\\{\frac{3}{2}}&{\frac{{ - 1}}{2}}&0\end{array}} \right]$

$\therefore$ $\frac{{A + {A^\prime }}}{2} + \frac{{A - {A^\prime }}}{2} = \left[ {\begin{array}{cccccccccccccccccccc}2&2&{\frac{5}{2}}\\2&{ - 1}&{\frac{3}{2}}\\{\frac{5}{2}}&{\frac{3}{2}}&2\end{array}} \right] + \left[ {\begin{array}{cccccccccccccccccccc}0&1&{\frac{{ - 3}}{2}}\\{ - 1}&0&{\frac{1}{2}}\\{\frac{3}{2}}&{\frac{{ - 1}}{2}}&0\end{array}} \right]$

which is the required expression.