Browse Questions

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

Toolbox:
• A square matrix A=[a$_{ij}$] is said to be skew symmetric if A'=-A that is $[a_{ij}]= -[a_{ji}]$ for all possible value of i and j.
To prove:
All diagonal elements of a skew-symmetric matrix are all zero.
Proof:
Let $A=[a_{ij}]_{n\times n}$ be a skew symmetric matrix.
$\Rightarrow a_{ij}=-a_{ji}$ for all i & j.
$\Rightarrow a_{ii}=-a_{ii}$ $\quad (Put \;j=i)$
$\Rightarrow 2a_{ii}=0\Rightarrow a_{ii}=0.$
Thus in a skew symmetric matrix all elements along the principal diagonal are zero.