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

Matrix A can be written as, $A=\frac{A+A^T}{2}+\frac{A-A^T}{2}$
Clearly first part is symmetric matrix and second part is skew symmetric matrix,

$\text{Symmetric part:}\left[\begin{array}{ccc}1& 4& 7/2\\ 4& 8& 3\\ 7/2& 3& 7\end{array}\right]$

$\text{and Skew-Symmetric part:}\left[\begin{array}{ccc}0& -2& 1/2\\ 2& 0& -2\\ -1/2& 2& 0\end{array}\right]$