Question

Show that the elements on the main diagonal of a skew-symmetric matrix are all zero.

Answer 1

Let $A=\left[a_{ij}\right]$ be a skew-symmetric matrix. Then,

$a_{ij}=-a_{ji}$ for all i, j.

$a_{ii}=-a_{ii}$ for all values of i.

$2a_{ii}=0$

$a_{ii}=0$ for all values of i.

$⟹a_{11}=a_{22}=a_{33}=...=a_{nn}=0$.