Let A $\epsilon$ $\mathcal M_n$. Define in $\mathbb {R^n}$x$\mathbb {R^n}$:
$\langle$x,y$\rangle_A$=$\langle$Ax,Ay$\rangle$
Prove that $\langle \rangle_A$ is an inner product in $\mathbb R^n$, if and only if, A is invertible. (In the right side on the equation, we have the canonical inner product in $\mathbb R^n$)
If $A$ is not invertible the null space of $A$ is not just $\{\vec{0}\}$. Therefore $<x, x>_A$ will be 0 for every element in the nullspace of $A$, which is not prohibited by definition of an inner product.