Answer
See below
Work Step by Step
Without the use of determinants, the solution to this problem appears almost instantly.
Since A is invertible, $A^{-1}$ exists. $B=C$ is obtained by multiplying $AB = AC$ by $A^{-1}$ from the left.
However, the problem is possibly mentioned incorrectly; what is supposed to be proved using determinants is that if $A$ is invertible, $B$ and $C$ are matrices with $AB = AC$, then $\det(B) = det (C)$.
Then, the solution can be showed by taking the determinants of both sides of $AB=AC$:
$\det(AB)=\det(AC)$
According to property P9:
$\det(A).\det(B)=\det(A)\det(C)$
Theorem 3.2.5 states that $\det(A) = 0$ since matrix $A$ is invertible. As a result, $\det (B) =\det (C)$
