linear algebra - Which of the following is false?

Let $A^{-1} = \begin{pmatrix}1& 2\\ 2& 3\end{pmatrix}$

Which of the following statements is FALSE?

A. For arbitrary 2x2 matrices, B, C, if BA = BC then A = C.

B. $A^T$ is invertible.

C. For arbitrary 2x2 matrices, B, C, if AB = AC then B = C.

D. $AA^{-1} = A^{-1}A$

E. $A$ is a symmetric matrix.

I know for sure that B, D, E are true, but I'm stuck up on A and C. The answer key tells me A is false, but I don't understand why. It seems like A and C are asking essentially the same thing. If the first two matrices are the same, then are the other two equivalent. What makes A different than C?

Answer

The key is you need to be able to apply inverses on both sides. For part C, we are already given $A$ is an invertible matrix, hence $AB = AC$ implies $A^{-1}AB= A^{-1}AC$ and $B=C$. However, in part A, we do not know that $B$ is invertible. In particular, take $B$ to be the zero matrix and $C$ a different matrix from $A$.