Question

Give examples of matrices

 A, B and C such that AB = AC but B ≠ C, A ≠ 0.

 

Answer 1

\[\left( iv \right) Let A = \begin{bmatrix}1 & 0 \\ 0 & 0\end{bmatrix} , B = \begin{bmatrix}0 & 0 \\ 0 & 1\end{bmatrix} \text{and C} = \begin{bmatrix}0 & 0 \\ 0 & 2\end{bmatrix}\]

\[ \therefore AB = \begin{bmatrix}1 & 0 \\ 0 & 0\end{bmatrix}\begin{bmatrix}0 & 0 \\ 0 & 1\end{bmatrix}\]

\[ \Rightarrow AB = \begin{bmatrix}0 + 0 & 0 + 0 \\ 0 + 0 & 0 + 0\end{bmatrix} = \begin{bmatrix}0 & 0 \\ 0 & 0\end{bmatrix}\]

\[and AC = \begin{bmatrix}1 & 0 \\ 0 & 0\end{bmatrix}\begin{bmatrix}0 & 0 \\ 0 & 2\end{bmatrix}\]

\[ \Rightarrow AC = \begin{bmatrix}0 + 0 & 0 + 0 \\ 0 + 0 & 0 + 0\end{bmatrix} = \begin{bmatrix}0 & 0 \\ 0 & 0\end{bmatrix}\]

\[\]

Thus,
AB = AC
But B ≠ C and A ≠ 0.