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Question
Give examples of matrices
A, B and C such that AB = AC but B ≠ C, A ≠ 0.
Solution
\[\left( iv \right) Let A = \begin{bmatrix}1 & 0 \\ 0 & 0\end{bmatrix} , B = \begin{bmatrix}0 & 0 \\ 0 & 1\end{bmatrix} \text{and C} = \begin{bmatrix}0 & 0 \\ 0 & 2\end{bmatrix}\]
\[ \therefore AB = \begin{bmatrix}1 & 0 \\ 0 & 0\end{bmatrix}\begin{bmatrix}0 & 0 \\ 0 & 1\end{bmatrix}\]
\[ \Rightarrow AB = \begin{bmatrix}0 + 0 & 0 + 0 \\ 0 + 0 & 0 + 0\end{bmatrix} = \begin{bmatrix}0 & 0 \\ 0 & 0\end{bmatrix}\]
\[and AC = \begin{bmatrix}1 & 0 \\ 0 & 0\end{bmatrix}\begin{bmatrix}0 & 0 \\ 0 & 2\end{bmatrix}\]
\[ \Rightarrow AC = \begin{bmatrix}0 + 0 & 0 + 0 \\ 0 + 0 & 0 + 0\end{bmatrix} = \begin{bmatrix}0 & 0 \\ 0 & 0\end{bmatrix}\]
\[\]
Thus,
AB = AC
But B ≠ C and A ≠ 0.
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