If the matrix AB is zero, then

It is not necessary that either $A=O$ or, $B=O$ $A =O$ or $B=O$ $A =O$ and $B=O$ all the above statements are wrong

It is not necessary that either $A=O$ or, $B=O$

If $A=\begin{bmatrix}1&0\\0&0\end{bmatrix},B=\begin{bmatrix}0&0\\0&1\end{bmatrix}$, then $AB=\begin{bmatrix}0&0\\0&0\end{bmatrix}$ However, $A≠O, B≠O$. So, option (1) is correct.