Inverse of orthogonal matrix is orthogonal matrix?

Question

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.

Answer 1

If $A^t = A^{-1}$, then taking inverses of both sides, we have $(A^{t})^{-1} = A = (A^t)^t$.

Answer 2

If $Q$ is orthogonal, i.e. $QQ^T = I = Q^TQ$, then $Q^T Q^{TT} = Q^TQ = I = QQ^T = Q^{TT}Q^T$. So, $Q^T$ is orthogonal. Because $Q^{-1} = Q^T$, it suffices to check that. And you're done.

Answer 3

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.