The doctor give us Question want to Prove that:
If A is Orthogonal (mxm) matrix and (I+A) is invertible THEN Prove --> $$ (I-A) (I+A)^{-1} $$ is an Skew-Symmetric matrix
My Question
1) how to Prove his Question ?....Thanks in Advance
2026-03-29 05:44:12.1774763052
If A is Orthogonal (mxm) matrix and (I+A) is invertible THEN Prove --> $ (I-A) (I+A)^{-1} $ is an Skew-Symmetric matrix
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$(I-A^\top)(I+A)=-A^\top+A=-(I+A^\top)(I-A)\Rightarrow (I+A^\top)^{-1}(I+A^\top)=- (I-A)(I+A)^{-1}. $
Because $(X^\top)^{-1}=(X^{-1})^\top$ and $X^\top Y^\top=(YX)^\top$ we have $(I+A^\top)^{-1}(I+A^\top)=[(I+A)(I+A)^{-1}]^\top$ and won.