Let $$A'$$ be transpose of $$A$$, $$A^{+}$$ be MP-inverse of $$A$$.

Prove that if $$A=A'$$, then $$A b=0$$ if and only if $$A^+b=0$$.

Proof:

$$A b=0$$ implies that $$b=(I-A^+A)q$$ for some $$q$$.  Then,

$$A^+ b \\ =A^+(I-A^+A)q \\ =A^+(I-A'^+A')q \\ =A^+(I-(AA^+)')q \\ =A^+(I-AA^+)q \\ =A^+q-A^+AA^+q \\ =A^+q-A^+q \\ =0$$

Because $$A^{++}=A$$, the proof is completed.