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The problem provides matrix \( A \) and an equation involving the inverse of matrix products to find matrix \( B \). Let's break it down:

- Matrix \( A \) is given as:

\[
A = \begin{pmatrix} 3 & -1 \\ -2 & 1 \end{pmatrix}
\]

- The equation is:

\[
\left(A \cdot B^{-1}\right)^{-1} = \begin{pmatrix} 2 & 1 \\ 4 & 3 \end{pmatrix}
\]

To solve for matrix \( B \), we can use the following steps:

1. Rewrite the equation:
   \[
   A \cdot B^{-1} = \left(\begin{pmatrix} 2 & 1 \\ 4 & 3 \end{pmatrix}\right)^{-1}
   \]

2. Calculate the inverse of the matrix \( \begin{pmatrix} 2 & 1 \\ 4 & 3 \end{pmatrix} \).

3. Multiply both sides of the equation by \( A^{-1} \) to solve for \( B^{-1} \), then find \( B \).

I'll begin with the calculations now.The matrix \( B \) is:

\[
B = \begin{pmatrix} 4 & -1 \\ 6 & -1 \end{pmatrix}
\]

Would you like more details or have any questions?

Here are some related questions you might explore:

1. How is the inverse of a matrix calculated?
2. What are the properties of matrix multiplication and inversion?
3. How can you verify the solution by multiplying matrices?
4. What happens if a matrix is singular (non-invertible)?
5. How does the determinant affect the invertibility of a matrix?

**Tip:** Always verify your result by substituting it back into the original equation to ensure correctness.

Jika A = \begin{pmatrix} 3 & -1 \\ -2 & 1 \end{pmatrix}, dan (A \cdot B^{-1})^{-1} = \begin{pmatrix} 2 & 1 \\ 4 & 3 \end{pmatrix}, maka Matriks B adalah?

This problem involves solving for Matrix B given matrix A and an inverse matrix equation. Learn how to apply matrix multiplication, matrix inverses, and properties of linear algebra to find the solution.

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