Answer
See below
Work Step by Step
Assume that $A=B=\begin{bmatrix}
0 &1\\0&0
\end{bmatrix}$
then $AB=\begin{bmatrix}
0 &1\\0&0
\end{bmatrix}\begin{bmatrix}
0 &1\\0&0
\end{bmatrix}=0\\
\rightarrow rank(AB)=0$
Since $rank(A)=rank (B)= 1\rightarrow rank(A).rank(B)=1$
Hence, the statement is false.