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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.

Solve the equation AB = BC for A, assuming that A, B, and C are square matrices and B is invertible.

Learn how to solve the matrix equation AB = BC for A, where A, B, and C are square matrices, and B is invertible. Detailed explanation of matrix inversion and algebra, suitable for undergraduate students studying linear algebra.

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