Symmetric, real, invertible matrix: How to prove component multiplication equals Kronecker?

Question

Im currently looking for a prove. Given a real and invertible matrix $M$ with $M=M^T$, I would like to prove $M_{ij} M_{jk}^{-1}=\delta_{ik}$, where $\delta_{ik}$ is the Kronecker-delta (defined by $\delta_{i=k}=1$ and $\delta_{i\neq k}=0$).

Answer 1

It is obviously false: any random real symmetric matrix will provide a counterexample. For example, take $M=\begin{pmatrix}2 & 1 \cr 1 & 1\end{pmatrix}$, so $M^{-1}=\begin{pmatrix}\phantom{-}1 & -1 \cr -1 & \phantom{-}2\end{pmatrix}$