We are given the matrix equation $AB = BC$, where $A$, $B$, and $C$ are square matrices, and $B$ is invertible. The goal is to solve for $A$.

Steps to solve for $A$:

1. Start with the equation: $AB = BC$
2. Since $B$ is invertible, we can multiply both sides of the equation on the right by $B^{-1}$ (the inverse of $B$). $AB B^{-1} = BC B^{-1}$
3. By matrix multiplication rules, $BB^{-1} = I$ (the identity matrix), so we simplify: $A = C B^{-1}$

Thus, the solution for $A$ is: $A = C B^{-1}$

Would you like further details on this solution or any clarification?

Here are five related questions to explore:

1. How does the invertibility of matrix $B$ affect the solution?
2. What properties of square matrices make it possible to solve this equation?
3. Can the equation be solved if $B$ is not invertible? Why or why not?
4. How do you calculate the inverse of a matrix in general?
5. Are there special cases where $A$ would have a unique form, depending on $C$ and $B$?

Tip: Always verify whether a matrix is invertible by checking if its determinant is non-zero.