Find two 3x3 matrices A and B each containing at max three zeros that satisfy the condition, rank($\textbf{A}$)=rank($\textbf{B}$)=rank($\textbf{AB}$)=rank($\textbf{BA}$)+1=2
My attempt: Let A=[ $a_{1}$ $a_{2}$ $a_{3}$] and B=[$b_{1}$ $b_{2}$ $b_{3}$]
Let {$a_{1}$, $a_{2}$} form a basis for Col A and {$b_{1}$, $b_{2}$} form a basis for Col B.
AB= A [$b_{1}$ $b_{2}$ $b_{3}$] = [A$b_{1}$ A$b_{2}$ A$b_{3}$]
BA =B [$a_{1}$ $a_{2}$ $a_{3}$] = [B$a_{1}$ B$a_{2}$ B$a_{3}$]
We want Rank (AB)=2 and rank (BA)=1, if $b_{3}$ $\in$ Nul A and A$b_{1}$, A$b_{2}$ are two linearly independent vectors, matrix AB satisfy the condition. I don't know how to tackle this question, how should one think? Any help / hints would be appreciated!