Question

Sum of two skew symmetric matrices is always matrix.

Answer 1

Let A and B be two skew symmetric matrices.
Then,
$A=-A^T$...()
$B=-B^T\dots \left(2\right)$
Adding equation (1) and (2)
$A+B=-A^T-B^T$
$\Rightarrow A+B=-(A^T+B^T)$
$\Rightarrow A+B=-(A+B{)}^T$
$[\because (A+B{)}^T=A^T+B^T]$
If $A^T=-A$ or $A=-A^T$ then A is called skew-symmetric matrix.
$\therefore A+B$ is also skew symmetric matrix.
Hence, Sum of two skew symmetric matrices is always skew-symmetric matrix.