Question

Express the matrix $A=\left[\begin{array}{ccc}4& 2& -1\\ 3& 5& 7\\ 1& -2& 1\end{array}\right]$ as the sum of symmetric and a skew-symmetric matrix.

Answer 1

The symmetric matrix is of type,$\left[\begin{array}{ccc}x& a& b\\ a& y& c\\ b& c& z\end{array}\right]$ and skew-symmetric matrix is of type $\left[\begin{array}{ccc}0& f& g\\ -f& 0& h\\ -g& -h& 0\end{array}\right]$

Let,

$\left[\begin{array}{ccc}4& 2& -1\\ 3& 5& 7\\ 1& -2& 1\end{array}\right]$ = $\left[\begin{array}{ccc}x& a& b\\ a& y& c\\ b& c& z\end{array}\right]$ + $\left[\begin{array}{ccc}0& f& g\\ -f& 0& h\\ -g& -h& 0\end{array}\right]$

Equating we get, $x=4,y=5,z=1$,$a=\frac{5}{2},f=\frac{-1}{2},b=0,g=-1,c=\frac{5}{2},h=\frac{9}{2}$

$\to$$\left[\begin{array}{ccc}4& 2& -1\\ 3& 5& 7\\ 1& -2& 1\end{array}\right]$ = $\left[\begin{array}{ccc}4& \frac{5}{2}& 0\\ \frac{5}{2}& 5& \frac{5}{2}\\ 0& \frac{5}{2}& 1\end{array}\right]$ + $\left[\begin{array}{ccc}0& \frac{-1}{2}& -1\\ \frac{1}{2}& 0& \frac{9}{2}\\ 1& -\frac{9}{2}& 0\end{array}\right]$