# For the matrix, verify that

Question:

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

(i) $\left(A+A^{\prime}\right)$ is a symmetric matrix

(ii) $\left(A-A^{\prime}\right)$ is a skew symmetric matrix

Solution:

$A^{\prime}=\left[\begin{array}{ll}1 & 6 \\ 5 & 7\end{array}\right]$

(i) $A+A^{\prime}$$=\left[\begin{array}{ll}1 & 5 \\ 6 & 7\end{array}\right]+\left[\begin{array}{ll}1 & 6 \\ 5 & 7\end{array}\right]=\left[\begin{array}{ll}2 & 11 \\ 11 & 14\end{array}\right] \therefore\left(A+A^{\prime}\right)^{\prime}=\left[\begin{array}{ll}2 & 11 \\ 11 & 14\end{array}\right]=A+A^{\prime} Hence, \left(A+A^{\prime}\right) is a symmetric matrix. (ii) A-A^{\prime}$$=\left[\begin{array}{ll}1 & 5 \\ 6 & 7\end{array}\right]-\left[\begin{array}{ll}1 & 6 \\ 5 & 7\end{array}\right]=\left[\begin{array}{rr}0 & -1 \\ 1 & 0\end{array}\right]$

$\left(A-A^{\prime}\right)$$=\left[\begin{array}{cc}0 & 1 \\ -1 & 0\end{array}\right]=-\left[\begin{array}{cc}0 & -1 \\ 1 & 0\end{array}\right]=-\left(A-A^{\prime}\right)$

Hence, $\left(A-A^{\prime}\right)$ is a skew-symmetric matrix.