Question
Give an example of matrices \(A\) and \(B\) such that \[ AB = O \quad \text{but} \quad BA \ne O. \]
Solution
Step 1: Take Matrices
\[ A = \begin{bmatrix} 1 & 0 \\ 0 & 0 \end{bmatrix}, \quad B = \begin{bmatrix} 0 & 0 \\ 1 & 0 \end{bmatrix} \]Step 2: Compute \(AB\)
\[ AB = \begin{bmatrix} 1 & 0 \\ 0 & 0 \end{bmatrix} \begin{bmatrix} 0 & 0 \\ 1 & 0 \end{bmatrix} = \begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix} = O \]Step 3: Compute \(BA\)
\[ BA = \begin{bmatrix} 0 & 0 \\ 1 & 0 \end{bmatrix} \begin{bmatrix} 1 & 0 \\ 0 & 0 \end{bmatrix} = \begin{bmatrix} 0 & 0 \\ 1 & 0 \end{bmatrix} \ne O \]Final Answer
\[
A =
\begin{bmatrix}
1 & 0 \\
0 & 0
\end{bmatrix}, \quad
B =
\begin{bmatrix}
0 & 0 \\
1 & 0
\end{bmatrix}
\]
\[
AB = O \quad \text{but} \quad BA \ne O
\]