https://file.aiquickdraw.com/mathbot/file/1729187006542kf79lrg2.jpg

The question asks to compute \( B^{-1}C \), where:

\[
B = \begin{bmatrix} 
1 & 1 & 2 \\
2 & 3 & 5 \\
3 & 3 & 7 
\end{bmatrix}, 
\quad
C = \begin{bmatrix} 
5 & 4 \\
13 & 11 \\
16 & 14 
\end{bmatrix}
\]

### Steps to Solve:

1. **Find \( B^{-1} \) (the inverse of matrix \( B \)):**
   - First, calculate the determinant of matrix \( B \).
   - If the determinant is non-zero, use the adjugate method or other methods (like Gaussian elimination) to find the inverse of \( B \).

2. **Multiply \( B^{-1} \) by matrix \( C \):**
   - After obtaining \( B^{-1} \), compute the matrix product \( B^{-1}C \).

Let's start by computing \( B^{-1} \). I'll calculate the inverse and the final product for you.The result of \( B^{-1}C \) is:

\[
B^{-1}C = \begin{bmatrix} 
1 & -1 \\
2 & 1 \\
1 & 2 
\end{bmatrix}
\]

Would you like further details on any of the steps, or have any questions about the solution?

Here are some related questions you might explore:
1. How do you calculate the determinant of a 3x3 matrix?
2. What are the conditions for a matrix to be invertible?
3. Can all matrices be inverted?
4. How would you solve this problem using Gaussian elimination?
5. How do matrix operations, like multiplication, work for different matrix dimensions?

**Tip:** The inverse of a matrix exists only if the determinant is non-zero!

Let B = [[1, 1, 2], [2, 3, 5], [3, 3, 7]], C = [[5, 4], [13, 11], [16, 14]], what is B^{-1}C?

Learn how to find the product B^{-1}C for matrices B and C. This problem involves calculating the inverse of a 3x3 matrix B and multiplying it with a 3x2 matrix C. Suitable for advanced high school or college students studying linear algebra.

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