Here is a counterexample.
I will use MATLAB notation for matrices, so ; ends each row of the matrix.
Let M be the 2 by 3 matrix M = [1 2 3;4 5 6]. Then A = [ 1.25 0 0.25;0 0 0;0.25 0 1.25].
MM' = MAM'= [14 32;32 77].
Here is a counterexample.
I will use MATLAB notation for matrices, so ; ends each row of the matrix.
Let M be the 2 by 3 matrix M = [1 2 3;4 5 6]. Then A = [ 1.25 0 0.25;0 0 0;0.25 0 1.25].
MM' = MAM'= [14 32;32 77].