Let $A$ be an invertible skew-symmetric $(2n \times 2n)$-matrix. Prove that $A^{-1}$ is also skew-symmetric. (You may assume that $(AB)^T = B^TA^T$).
I did this with a $2 \times 2$ matrix and got that it worked, but I don't know how to show it for a general $2n \times 2n$ matrix, as it is a little harder to calculate the inverse of that. Obviously the hint comes into play somehow but I can't see how.
I have the definition of a skew symmetric bileanr function to be $B(u,v) = – B(v,u)$, but again, I can't see how to put this into matrix form and use that.
Can someone give me some hints please?
Best Answer
$(A^T)^{-1}=(A^{-1})^T$ and according to Wikipedia, a skew-symmetric matrix is a matrix that satisfies the condition $A^T=-A$. So $(A^{-1})^T=(A^T)^{-1}=(-A)^{-1}=-A^{-1}$ Why do you need $2n\times 2n$ condition?