let $A,B$ matrices $n \times n$ such that $A \ne B $ and $AB=0$ and $A\ne 0 , B\ne0$ prove $\left|A\right|^2+\left|B\right|^2=0$
My attempt:
$$AB=0 \implies \det(AB)=0 \implies \det(A)*\det(B)=0 \iff $$ $$|B|=0\qquad\text{or}\qquad |A|=0\qquad\text{or}\qquad|A|=|B|=0.$$
if $|A|=0 , |B| \ne0$ then $B^{-1}$ exist $\implies ABB^{-1}=0 \implies A=0 \implies$contradiction
if $|B|=0 , |A| \ne0$ same as before
and if $|B|=0 , |A| =0$ then also $\left|A\right|^2+\left|B\right|^2=0$
My question is does this correct what I did and if not where did I did wrong ?
thanks