Question

If $A$ is $n$ squared matrix, then prove that $A+A^{\prime }$ is symmetric and $A-A^{\prime }$ is skew symmetric.

Answer 1

For any matrix to be symmetric $A^{\prime }=A$ and to be skew symmetric $A^{\prime }=-A$
Now $(A+A^{\prime }{)}^{\prime }=A^{\prime }+(A^{\prime }{)}^{\prime }=A^{\prime }+A=A+A^{\prime }$
The condiction $A^{\prime }=A$ is satisfied by the matrix $A+A^{\prime }$ which therefore is symmetric.
Again $(A-A^{\prime }{)}^{\prime }=A^{\prime }-(A^{\prime }{)}^{\prime }=A^{\prime }-A=-(A-A^{\prime })$
The condition of $A^{\prime }=-A$ is satisfied by the matrix $A-A^{\prime }$ which is skew symmetric.