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10 Οκτ 2013 (πριν από 4 χρόνια και 1 μήνα)

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If $A$ is an invertible skew-symmetric matrix, then prove $A^{-1}$ is also skew symmetric
Asked by
on 2013-04-21T08:42:26-04:00
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
Answer by
on 2013-04-21T08:50:00-04:00
$(A^T)^{-1}=(A^{-1})^T$ 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?

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