If A is an invertible matrix, then A^{-1} exists, and it is such that A A^{-1}=A^{-1}A=I, where I is the identity matrix. In this case, from AB=AC, we could multiply both sides for A^{-1} to the left, and obtain A^{-1}AB=A^{-1}AC, which means B=C. So, if A is invertible, your statement cannot be proved.
So, A must surely be not invertible (i.e. its determinant must be zero). The simplest matrix can be the null matrix (every coefficient is zero). If you choose it as A, you'll obtain that AB=AC means 0=0 (where 0 is the zero matrix), regardless of B and C.
