Question

Prove that the diagonal elements of a skew-symmetric matrix are all zeroes.

Answer 1

Let A be a skew-symmetric matrix.

Then, by definition $A^{\prime }=-A$

The $(i,j)th$ element of $A^{\prime }$ = the $(i,j)th$ element of $(-A)$.

$⟹$ The $(j,i)th$ element of $A$ = $-$ the $(i,j)th$ element of $A$

For the diagonal elements, $i=j$

Therefore, the $(i,i)th$ element of $A$ = $-$ the $(i,i)th$ element of $A$

$⟹$ the $(i,i)th$ element of $A=0$

Hence, the diagonal elements of a skew-symmetric matrix are all zero.