If \(A=\begin{bmatrix}1&0\\0&0\end{bmatrix},B=\begin{bmatrix}0&0\\3&0\end{bmatrix}\) then
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → \(AB=0, BA\ne 0\)
\( AB = \begin{bmatrix}1&0\\0&0\end{bmatrix}\begin{bmatrix}0&0\\3&0\end{bmatrix} = \begin{bmatrix}0&0\\0&0\end{bmatrix} = O \)
\( BA = \begin{bmatrix}0&0\\3&0\end{bmatrix}\begin{bmatrix}1&0\\0&0\end{bmatrix} = \begin{bmatrix}0&0\\3&0\end{bmatrix} \ne O \)
So AB = O but BA ≠ O: matrix multiplication is not commutative, and AB = O does not force BA = O.
Why the other options are incorrect
Option (2) \(AB = 0, BA = 0\) — Incorrect. BA has the entry 3 in row 2, column 1, so BA ≠ O.
Option (3) \(A^2\ne A\) — Incorrect. \(A^2 = \begin{bmatrix}1&0\\0&0\end{bmatrix} = A\).
Option (4) \(B^2\ne 0\) — Incorrect. \(B^2 = \begin{bmatrix}0&0\\0&0\end{bmatrix} = O\).