Question

For the matrix $A=\left[\begin{array}{cc}1& 5\\ 6& 7\end{array}\right],$ verify that
$\left(i\right)$ $(A+A^{\prime })$ is a symmetric matrix.
$\left(ii\right)$ $(A-A^{\prime })$ is a skew symmetric matrix.

Answer 1

$\left(i\right)$ Given : $A=\left[\begin{array}{cc}1& 5\\ 6& 7\end{array}\right]$
$A^{\prime }=\left[\begin{array}{cc}1& 6\\ 5& 7\end{array}\right]$
$A+A^{\prime }=\left[\begin{array}{cc}1& 5\\ 6& 7\end{array}\right]+\left[\begin{array}{cc}1& 6\\ 5& 7\end{array}\right]$
$\Rightarrow A+A^{\prime }=\left[\begin{array}{cc}2& 11\\ 11& 14\end{array}\right]$
$\Rightarrow (A+A^{\prime }{)}^{\prime }=\left[\begin{array}{cc}2& 11\\ 11& 14\end{array}\right]$
$\because (A+A^{\prime }{)}^{\prime }=A+A^{\prime }$
Hence, $(A+A^{\prime })$ is a symmetric matrix.

$\left(ii\right)$ Given : $A=\left[\begin{array}{cc}1& 5\\ 6& 7\end{array}\right]$
$A^{\prime }=\left[\begin{array}{cc}1& 6\\ 5& 7\end{array}\right]$
$A-A^{\prime }=\left[\begin{array}{cc}1& 5\\ 6& 7\end{array}\right]-\left[\begin{array}{cc}1& 6\\ 5& 7\end{array}\right]$
$\Rightarrow A-A^{\prime }=\left[\begin{array}{cc}0& -1\\ 1& 0\end{array}\right]$
​​​​​​​$\Rightarrow (A-A^{\prime }{)}^{\prime }=\left[\begin{array}{cc}0& 1\\ -1& 0\end{array}\right]=-\left[\begin{array}{cc}0& -1\\ 1& 0\end{array}\right]$
$\Rightarrow (A-A^{\prime }{)}^{\prime }=-(A-A^{\prime })$
Hence, $(A-A^{\prime })$ is a skew symmetric matrix.