Give an example of matrices A, B and C, such that $AB = AC,$where A is non-zero matrix but $B \ne C$.

Let $A = \left[ {\begin{array}{llllllllllllllllllll}1&0\\0&0\end{array}} \right],B = \left[ {\begin{array}{llllllllllllllllllll}2&3\\4&0\end{array}} \right]$ and $C$

$= \left[ {\begin{array}{llllllllllllllllllll}2&3\\4&4\end{array}} \right]$

$\therefore$ $AB = \left[ {\begin{array}{llllllllllllllllllll}1&0\\0&0\end{array}} \right]\left[ {\begin{array}{llllllllllllllllllll}2&3\\4&0\end{array}} \right] = \left[ {\begin{array}{llllllllllllllllllll}2&3\\0&0\end{array}} \right]$

…..(i)
and $AC = \left[ {\begin{array}{llllllllllllllllllll}1&0\\0&0\end{array}} \right] \cdot \left[ {\begin{array}{llllllllllllllllllll}2&3\\4&4\end{array}} \right] = \left[ {\begin{array}{llllllllllllllllllll}2&3\\0&0\end{array}} \right]$

……(ii)
Thus, we see that $AB = AC$ [using Eqs.

(i) and (ii)]
where, A is non-zero matrix but $B \ne C$.