For a symmetric matrix $X$, is $A^T X^{-1} A$ symmetric for any $A$?

Question

This should be a straightforward answer but my matrix algebra skills are weak.

If I have a symmetric matrix $X$, is

$A^TX^{-1}A$ symmetric for any matrix $A$?

I know the inverse of a symmetric matrix is symmetric, but I'm having difficulty checking the rest of the conditions.

Answer 1

Hint: A matrix is symmetric if it is equal to it's transpose. The transpose of a product of matrices is the product of the transposes in the opposite order. So the transpose of $A^TX^{-1}A$ is:

$$(A^TX^{-1}A)^T=A^T(X^{-1})^T(A^T)^T=?$$