Proving Orthogonality: Simple Matrix Question with Skew-Symmetric A"

[the4thamigo_uk]

Homework Statement

Prove that if A is skew-symmetric (i.e. A = -A') then


(I - A) . inv(I + A)

is orthogonal, assuming that (I + A) is singular. inv(X) denotes inverse matrix of X. X' denotes transpose of X.

The Attempt at a Solution

I need to prove orthogonality :

I know for square matrices : inv(XY) = inv(Y) . inv(X) and (XY)' = Y'X'

So if X and Y were orthogonal then XY would be.

But in this case X = I - A and Y = inv(I + A), but are they orthogonal?

I also know that (I - A)' = (I + A)

But can't figure it out. Do I have to resort to the defn of inv(X) = adj(X) / det(X) ?


Would appreciate any clues... Thanks in advance

 

[the4thamigo_uk]

Aha stupid me... I now note that inv(X)' = inv(X')