TITLE: Inverse of orthogonal matrix is orthogonal matrix?
QUESTION [11 upvotes]: Is inverse of an orthogonal matrix an orthogonal matrix? I know its inverse is equal to its transpose, but I don't see where the orthogonality would come from.

REPLY [3 votes]: If $(\ ,\ )$ is an inner product on ${\bf R}^n$, then a matrix $A$ is orthogonal if $$ (Ax,Ax)=(x,x),\ \forall x\in {\bf R}^n $$
Note that $A$ has no kernel. Hence any $x$ has $y$ with $$ Ay=x$$
Then $$   (A^{-1}x,A^{-1}x)= (A^{-1}Ay,A^{-1}Ay)=(y,y)=(Ax,Ax)=(x,x)$$
Hence $A^{-1}$ is orthogonal.