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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]. MM' = MA*M'= [14 32;32 77].