Question

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

Answer 1

The statement is true for some non-invertible matrices `A`

Explanation:

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`.