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 = [ 1.2500 0 0.2500;0 0 0;0.2500 0 1.2500].
M*M' = M*A*M'= [14 32;32 77].