Question

Find the symmetric and skew symmetric parts of the matrix
$A=\left[\begin{array}{ccc}1& 2& 4\\ 6& 8& 1\\ 3& 5& 7\end{array}\right]$

Answer 1

$A^{\prime }=\left[\begin{array}{ccc}1& 6& 3\\ 2& 8& 5\\ 4& 1& 7\end{array}\right]$
Hence the symmetric and skew symmetric parts are
$\frac{1}{2}(A+A^{\prime })=\left[\begin{array}{ccc}1& 4& 7/2\\ 4& 8& 3\\ 7/2& 3& 7\end{array}\right]$ which is symmetric and
$\frac{1}{2}(A-A^{\prime })=\left[\begin{array}{ccc}0& -2& 1/2\\ 2& 0& -2\\ -1/2& 2& 0\end{array}\right]$ which is skew symmetric
Clearly the given matrix is the sum of the above two matrices.