Question
Give an example of matrices \(A, B, C\) such that \[ AB = AC \quad \text{but} \quad B \ne C,\ A \ne O. \]
Solution
Step 1: Take Matrices
\[ A = \begin{bmatrix} 1 & 1 \\ 1 & 1 \end{bmatrix}, \quad B = \begin{bmatrix} 1 & 0 \\ 0 & 0 \end{bmatrix}, \quad C = \begin{bmatrix} 0 & 0 \\ 1 & 0 \end{bmatrix} \]Step 2: Compute \(AB\)
\[ AB = \begin{bmatrix} 1 & 1 \\ 1 & 1 \end{bmatrix} \begin{bmatrix} 1 & 0 \\ 0 & 0 \end{bmatrix} = \begin{bmatrix} 1 & 0 \\ 1 & 0 \end{bmatrix} \]Step 3: Compute \(AC\)
\[ AC = \begin{bmatrix} 1 & 1 \\ 1 & 1 \end{bmatrix} \begin{bmatrix} 0 & 0 \\ 1 & 0 \end{bmatrix} = \begin{bmatrix} 1 & 0 \\ 1 & 0 \end{bmatrix} \]Step 4: Compare
\[ AB = AC \quad \text{but} \quad B \ne C,\ A \ne O \]Final Answer
\[
A =
\begin{bmatrix}
1 & 1 \\
1 & 1
\end{bmatrix}, \quad
B \ne C,\quad AB = AC
\]