Question

For the matrix, $A=\left[\begin{array}{cc}1& 5\\ 6& 7\end{array}\right]$, verify that

(i) (A+A') is a symmetric matrix.

(ii) (A-A') is a skew-symmetric matrix.

Answer 1

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

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