Question 1201590
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(A-B)*(A+B) = A*A - B*A + A*B - B*B = (A^2 - B^2) + (A*B - B*A).


From this identity, which is valid for all matrices ALWAYS,  you can easily conclude that  

    (A-B)*(A+B) = A^2 - B^2  if and only if  A*B - B*A = 0.


It is the same as to say that 

    (A-B)*(A+B) = A^2 - B^2  if and only if  A*B = B*A.
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Solved.