TITLE: Is the inverse of a symmetric matrix also symmetric?
QUESTION [131 upvotes]: 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.

REPLY [2 votes]: Inverse of $A$ can be expressed as a polynomial $p(A)$ of $A$ (from Cayley-Hamilton theorem).
So it is sufficient to prove that if $A$ is symmetric then power $A^k$ is symmetric, sum of symmetric matrices is symmetric and multiply by scalar is symmetric.