If $A=\begin{bmatrix}1&0\\0&0\end{bmatrix},B=\begin{bmatrix}0&0\\3&0\end{bmatrix}$ then

$AB=0, BA≠0$

The correct answer is Option (1) → $AB≠0, BA≠0$

Given matrices:

$A = \begin{bmatrix}1 & 0 \\ 0 & 0\end{bmatrix}$, $B = \begin{bmatrix}0 & 0 \\ 3 & 0\end{bmatrix}$

Compute $AB$:

$AB = \begin{bmatrix}1 & 0 \\ 0 & 0\end{bmatrix} \begin{bmatrix}0 & 0 \\ 3 & 0\end{bmatrix} = \begin{bmatrix}0 & 0 \\ 0 & 0\end{bmatrix} = 0$

Compute $BA$:

$BA = \begin{bmatrix}0 & 0 \\ 3 & 0\end{bmatrix} \begin{bmatrix}1 & 0 \\ 0 & 0\end{bmatrix} = \begin{bmatrix}0 & 0 \\ 3 & 0\end{bmatrix} = B \neq 0$

Correct statements: $AB = 0, BA \neq 0$