A and B are 3x3 matrices. Show that $AB \neq 0$ if both matrices have a rank of 2.

Question

A and B are $3\times 3$ matrices. I want to show that $AB \neq 0$ if both matrices have rank 2.

Answer 1

Hint: Use Sylvester's inequality \begin{align} \operatorname{Rank}(AB) \geq \operatorname{Rank}(A)+\operatorname{Rank}(B)-3 \end{align}