https://file.aiquickdraw.com/mathbot/file/172686137671737zfmfdz.jpg

The given equation is:

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

with the matrices:

\[
A = \begin{pmatrix} -1 & -3 \\ 2 & 0 \end{pmatrix}, \quad B = \begin{pmatrix} -2 & 4 \\ 0 & 6 \end{pmatrix}, \quad C = \begin{pmatrix} 2 & 0 \\ 10 & -1 \end{pmatrix}.
\]

The task is to find \( X \).

### Steps:
1. First, find the inverses \( B^{-1}, A^{-1}, C^{-1} \).
2. Then solve for \( X \) by rearranging the equation: 
   \[
   X = B C^{-1} A
   \]
   (multiplying both sides by \( B \) and \( A \)).

I will calculate these now.Here are the results of the calculations:

1. The inverse of matrix \( A \) is:
   \[
   A^{-1} = \begin{pmatrix} 0 & 0.5 \\ -0.3333 & -0.1667 \end{pmatrix}
   \]

2. The inverse of matrix \( B \) is:
   \[
   B^{-1} = \begin{pmatrix} -0.5 & 0.3333 \\ 0 & 0.1667 \end{pmatrix}
   \]

3. The inverse of matrix \( C \) is:
   \[
   C^{-1} = \begin{pmatrix} 0.5 & 0 \\ 5 & -1 \end{pmatrix}
   \]

4. The matrix \( X \) is:
   \[
   X = \begin{pmatrix} -27 & -57 \\ -42 & -90 \end{pmatrix}
   \]

Would you like more details or explanations on any step?

Here are 5 questions related to this problem:
1. What are the properties of matrix inverses?
2. How do you compute the determinant of a 2x2 matrix?
3. Can every matrix be inverted? Why or why not?
4. What is the significance of the inverse of a matrix in linear algebra?
5. How would the solution change if matrix \( B \) were not invertible?

**Tip:** To verify if a matrix is invertible, its determinant should not be zero.

Solve the equation B^{-1} X A^{-1} = C^{-1}, given matrices A = [-1, -3; 2, 0], B = [-2, 4; 0, 6], and C = [2, 0; 10, -1]. Find matrix X.

Solve the equation B^{-1} X A^{-1} = C^{-1}, where A, B, and C are 2x2 matrices. Learn how to compute matrix inverses and use matrix multiplication to find X. Detailed step-by-step solution provided.

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