If A,B, C are three non-Zero …

Let {tex}A = \left[ {\begin{array}{*{20}{c}} 1&0 \\ 0&0 \end{array}} \right],B = \left[ {\begin{array}{*{20}{c}} 2&3 \\ 4&0 \end{array}} \right]{/tex} and {tex}C = \left[ {\begin{array}{*{20}{c}} 2&3 \\ 4&4 \end{array}} \right]\left[ {\because B \ne C} \right]{/tex}
{tex}\therefore AB = \left[ {\begin{array}{*{20}{c}} 1&0 \\ 0&0 \end{array}} \right]\left[ {\begin{array}{*{20}{c}} 2&3 \\ 4&0 \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} 2&3 \\ 0&0 \end{array}} \right]{/tex} ....(i)
And {tex}\therefore AC = \left[ {\begin{array}{*{20}{c}} 1&0 \\ 0&0 \end{array}} \right] \cdot \left[ {\begin{array}{*{20}{c}} 2&3 \\ 4&0 \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} 2&3 \\ 0&0 \end{array}} \right]{/tex} .....(ii)
Thus, we see that AB = AC [using Eqs. (i) and (ii)]
Where, A is non-zero matrix but {tex}B \ne C{/tex}