Question

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

Answer 1

Let A be a skew-symmetric matrix. Then by def., $A^T=-A$

So, the $(i,j{)}^{th}$ element of $A^T=the(i,j{)}^{th}$ element of (-A)

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

For the diagonal elements, i=j then, the $(i,i{)}^{th}$ element of A=-the $(i,i{)}^{th}$ element of A

$\Rightarrow 2\times the(i,i{)}^{th}$ element of A=0 $\therefore$ the $(i,i{)}^{th}$ element of A=0

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