Is the inverse of a symmetric matrix also symmetric?

Let $A$ be a symmetric invertible matrix, $A^T=A$, $A^{-1}A = A A^{-1} = I$ Can it be shown that $A^{-1}$ is also symmetric?

I seem to remember a proof similar to this from my linear algebra class, but it has been a long time, and I can’t find it in my text book.

You can’t use the thing you want to prove in the proof itself, so the above answers are missing some steps. Here is a more complete proof. Given A is nonsingular and symmetric, show that $A^{-1} = (A^{-1})^T$:

since $AA^{-1} = I$,

since $(AB)^T = B^TA^T$,

since $AA^{-1} = A^{-1}A = I$, we rearrange the left side

since $A = A^T$, we substitute the right side

and we are done.