# For matrix multiplication, how do I prove that if AB=AC, B may not equal C?

The statement is true for some non-invertible matrices $A$
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 $A B = A C$, we could multiply both sides for ${A}^{- 1}$ to the left, and obtain ${A}^{- 1} A B = {A}^{- 1} A C$, 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 $A B = A C$ means $0 = 0$ (where $0$ is the zero matrix), regardless of $B$ and $C$.