# Rqgg Page 1/1

Ηλεκτρονική - Συσκευές

10 Οκτ 2013 (πριν από 3 χρόνια και 6 μήνες)

63 εμφανίσεις

If $A$ is an invertible skew-symmetric matrix, then prove $A^{-1}$ is also skew symmetric
Kaish
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?

Antoine
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?

Archives
Science
Mathematics
Apr 21st, 2013
Week 16, 2013
April, 2013
Related
Linear Algebra Question 4
Relations between Linear Transformations and their Matrices
Diagonalizing Matrices
Linear Algebra, Unitary matrices
When do two matrices have the same column space?
Describe all matrices similar to a certain matrix.
Relationship betweeen commutativity of linear maps and their matrices
Relation between positive definite Hermitian matrices with their inverses
Relation between positive define Hermitian matrices with its inverses
question on linear algebra-matrices
View Online
http://www.rqgg.net/topic/nrqzy-if-a-is-an-invertible-skew-symmetric-matrix-then-prove-a-1-is-also-sk
ew-symmetric.html
Rqgg
Page 1/1