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The problem asks to find matrix \( C \) given the following information:

1. \( BC = \begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{bmatrix} \),
2. \( B^{-1} = \begin{bmatrix} 1 & 2 & 3 \\ 0 & 1 & 4 \\ 0 & 0 & 1 \end{bmatrix} \).

### Step-by-Step Solution:

To find matrix \( C \), we use the equation:

\[
BC = A
\]
where \( A = \begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{bmatrix} \).

To isolate \( C \), we multiply both sides of the equation by \( B^{-1} \) from the left:

\[
B^{-1} BC = B^{-1} A
\]

Since \( B^{-1}B = I \) (the identity matrix), we are left with:

\[
C = B^{-1} A
\]

Now, let's compute \( C \) by multiplying \( B^{-1} \) with \( A \):

\[
B^{-1} = \begin{bmatrix} 1 & 2 & 3 \\ 0 & 1 & 4 \\ 0 & 0 & 1 \end{bmatrix}, \quad
A = \begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{bmatrix}
\]

Multiplying these two matrices:

\[
C = \begin{bmatrix} 1 & 2 & 3 \\ 0 & 1 & 4 \\ 0 & 0 & 1 \end{bmatrix}
\begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{bmatrix}
\]

### Matrix multiplication:
1. First row of \( B^{-1} \) with each column of \( A \):
   - \( (1 \cdot 1) + (2 \cdot 4) + (3 \cdot 7) = 1 + 8 + 21 = 30 \)
   - \( (1 \cdot 2) + (2 \cdot 5) + (3 \cdot 8) = 2 + 10 + 24 = 36 \)
   - \( (1 \cdot 3) + (2 \cdot 6) + (3 \cdot 9) = 3 + 12 + 27 = 42 \)

2. Second row of \( B^{-1} \) with each column of \( A \):
   - \( (0 \cdot 1) + (1 \cdot 4) + (4 \cdot 7) = 0 + 4 + 28 = 32 \)
   - \( (0 \cdot 2) + (1 \cdot 5) + (4 \cdot 8) = 0 + 5 + 32 = 37 \)
   - \( (0 \cdot 3) + (1 \cdot 6) + (4 \cdot 9) = 0 + 6 + 36 = 42 \)

3. Third row of \( B^{-1} \) with each column of \( A \):
   - \( (0 \cdot 1) + (0 \cdot 4) + (1 \cdot 7) = 0 + 0 + 7 = 7 \)
   - \( (0 \cdot 2) + (0 \cdot 5) + (1 \cdot 8) = 0 + 0 + 8 = 8 \)
   - \( (0 \cdot 3) + (0 \cdot 6) + (1 \cdot 9) = 0 + 0 + 9 = 9 \)

So, the matrix \( C \) is:

\[
C = \begin{bmatrix} 30 & 36 & 42 \\ 32 & 37 & 42 \\ 7 & 8 & 9 \end{bmatrix}
\]

### Final Answer:
\[
C = \begin{bmatrix} 30 & 36 & 42 \\ 32 & 37 & 42 \\ 7 & 8 & 9 \end{bmatrix}
\]

Would you like further details on the steps?

Here are 5 follow-up questions:
1. What are the properties of an invertible matrix?
2. How do you calculate the inverse of a 3x3 matrix manually?
3. Why does multiplying a matrix by its inverse yield the identity matrix?
4. What are the applications of matrix inversion in solving linear systems?
5. How can we verify if the multiplication between two matrices is correct?

**Tip:** Always double-check your matrix multiplication to avoid simple errors in calculations.

Find the matrix C given BC and the inverse of matrix B.

Learn how to find matrix C when given the product BC and the inverse of matrix B. This detailed solution involves matrix multiplication and the properties of the inverse matrix theorem, making it ideal for students studying linear algebra.

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