# How do I prove that an anti-symmetric matrix $A$ is not invertible? [duplicate]

$A$ is a square anti symmetric matrix with dimension $n\times n$.

It is known that $n$ is an odd number. Prove that $A$ is not invertible.

How do I prove this? any hints please?

## marked as duplicate by user26857, user228113, Jeremy Rickard, Takumi Murayama, pjs36Apr 24 '16 at 0:33

$$\det(A)=\det(A^T)=\det(-A)=(-1)^n\det(A)=-\det(A)$$ since $n$ is odd. hence $$\det(A)=0$$
• I did not get it, why $$\det(A)=\det(A^T)$$ and why $$\det(A^T)=\det(-A)$$ – LiziPizi Nov 29 '15 at 17:10
• antisymmetric means $A^T=-A$, by definition – Kushal Bhuyan Nov 29 '15 at 17:13