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Mathematics Question on Matrices

For the matrix A=[15\67]\begin{bmatrix}1&5\\\6&7\end{bmatrix},verify that
I. (A+A') is a symmetric matrix
II. (A-A') is a skew symmetric matrix

Answer

A'= [16\57]\begin{bmatrix}1&6\\\5&7\end{bmatrix}

(i)A+A'=[15\67]\begin{bmatrix}1&5\\\6&7\end{bmatrix}+[16\57]\begin{bmatrix}1&6\\\5&7\end{bmatrix}

=[211\1114]\begin{bmatrix}2&11\\\11&14\end{bmatrix}

therefore (A+A')'= [211\1114]\begin{bmatrix}2&11\\\11&14\end{bmatrix}=A+A'
Hence,(A+A') is a symmetric matrix.

(ii)A-A'= [15\67]\begin{bmatrix}1&5\\\6&7\end{bmatrix}-[16\57]\begin{bmatrix}1&6\\\5&7\end{bmatrix}

=[0−1\10]\begin{bmatrix}0&-1\\\1&0\end{bmatrix}

(A-A')'=[01−10]\begin{bmatrix}0&1\\\\-1&0\end{bmatrix}=-[0−1\10]\begin{bmatrix}0&-1\\\1&0\end{bmatrix}=-(A-A')

Hence,(A-A') is a skew-symmetric matrix.