Give an example of matrices A, B and C such that AB = AC, where A is non-zero
matrix, but B ≠ C.
Let $A=\left[\begin{array}{ll}1 & 0 \\ 0 & 0\end{array}\right], B=\left[\begin{array}{ll}1 & 1 \\ 1 & 0\end{array}\right]$ and $C=\left[\begin{array}{ll}1 & 1 \\ 1 & 2\end{array}\right] \quad[\because B \neq C]$
$A B=\left[\begin{array}{ll}1 & 0 \\ 0 & 0\end{array}\right]\left[\begin{array}{ll}1 & 1 \\ 1 & 0\end{array}\right]=\left[\begin{array}{ll}1 & 1 \\ 0 & 0\end{array}\right]$ $\ldots$ (i)
and $A C=\left[\begin{array}{ll}1 & 0 \\ 0 & 0\end{array}\right]\left[\begin{array}{ll}1 & 1 \\ 1 & 2\end{array}\right]=\left[\begin{array}{ll}1 & 1 \\ 0 & 0\end{array}\right]$ $\ldots$ (ii)
From (i) and (ii),
Hence, AB = AC but B ≠ C.
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