Show that this statement is false: if $M$ is positive definite and $AMA^T = M$, then $A$ is orthogonal

Question

So the question is: Show that the following statement is false; if $M$ is positive definite, and $A$ a real $N\times N$ matrix, and $AMA^T = M$ then $A$ is orthogonal.

After a lot of fail and error, I still haven't found the solution. I hope though that someone can help me out on this one.

Answer 1

Let $M$ be a diagonal matrix with distinct positive diagonal entries and let $Q$ be a real orthogonal matrix that is not diagonal. Set $A=M^{1/2}QM^{-1/2}$, where $M^{1/2}$ and $M^{1/2}$ are computed entrywise in the diagonal. Show that $AMA^T=M$ but $AA^T\neq I$.

Answer 2

Let $M$ be the 2 by 2 matrix, diagonal, with diagonal entries 1,2. Find all $A.$