Question

Express $A=\left[\begin{array}{ccc}2& 5& -1\\ 3& 1& 5\\ 7& 6& 9\end{array}\right]$as sum of symmetric and skew- symmetric matrices.

Answer 1

Let $A=\left[\begin{array}{ccc}2& 5& -1\\ 3& 1& 5\\ 7& 6& 9\end{array}\right]$

Therefore,

$A^{\prime }=\left[\begin{array}{ccc}2& 3& 7\\ 5& 1& 6\\ -1& 5& 9\end{array}\right]$

Now,

$A+A^{\prime }=\left[\begin{array}{ccc}2& 5& -1\\ 3& 1& 5\\ 7& 6& 9\end{array}\right]+\left[\begin{array}{ccc}2& 3& 7\\ 5& 1& 6\\ -1& 5& 9\end{array}\right]=\left[\begin{array}{ccc}4& 8& 6\\ 8& 2& 11\\ 6& 11& 18\end{array}\right]$

$\Rightarrow {(A+A^{\prime })}^{\prime }=\left[\begin{array}{ccc}4& 8& 6\\ 8& 2& 11\\ 6& 11& 18\end{array}\right]=(A+A^{\prime })\Rightarrow$ Symmetric matrix

Similarly,

$A-A^{\prime }=\left[\begin{array}{ccc}2& 5& -1\\ 3& 1& 5\\ 7& 6& 9\end{array}\right]-\left[\begin{array}{ccc}2& 3& 7\\ 5& 1& 6\\ -1& 5& 9\end{array}\right]=\left[\begin{array}{ccc}0& 2& -8\\ -2& 0& -1\\ 8& 1& 0\end{array}\right]$

$\Rightarrow {(A-A^{\prime })}^{\prime }=\left[\begin{array}{ccc}0& -2& 8\\ 2& 0& 1\\ -8& -1& 0\end{array}\right]=-(A-A^{\prime })\Rightarrow$ Skew symmetric matrix

Now,

$\therefore A=\underset{\left(\text{Symmetric matrix}\right)}{\frac{1}{2}(A+A)}+\underset{\left(\text{Skew - symmetric matrix}\right)}{\frac{1}{2}(A-A^{\prime })}$