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The problem asks to find the matrix \( X \) given the equation \( A \cdot X = B \), where the matrices \( A \) and \( B \) are:

\[
A = \begin{pmatrix} 2 & 4 \\ 1 & 3 \end{pmatrix}, \quad B = \begin{pmatrix} 8 & 2 \\ 3 & 2 \end{pmatrix}.
\]

### Solution:
We can solve for \( X \) by finding the inverse of matrix \( A \) (denoted as \( A^{-1} \)) and then multiplying both sides of the equation by \( A^{-1} \):

\[
A^{-1} \cdot A \cdot X = A^{-1} \cdot B \quad \Rightarrow \quad X = A^{-1} \cdot B.
\]

#### Step 1: Find \( A^{-1} \)

The inverse of a 2x2 matrix \( A = \begin{pmatrix} a & b \\ c & d \end{pmatrix} \) is given by the formula:

\[
A^{-1} = \frac{1}{\text{det}(A)} \begin{pmatrix} d & -b \\ -c & a \end{pmatrix},
\]

where \( \text{det}(A) = ad - bc \) is the determinant of \( A \).

For the matrix \( A = \begin{pmatrix} 2 & 4 \\ 1 & 3 \end{pmatrix} \):
- \( a = 2, b = 4, c = 1, d = 3 \),
- \( \text{det}(A) = (2)(3) - (4)(1) = 6 - 4 = 2 \).

Thus, the inverse of \( A \) is:

\[
A^{-1} = \frac{1}{2} \begin{pmatrix} 3 & -4 \\ -1 & 2 \end{pmatrix} = \begin{pmatrix} \frac{3}{2} & -2 \\ -\frac{1}{2} & 1 \end{pmatrix}.
\]

#### Step 2: Multiply \( A^{-1} \) by \( B \)

Now, we multiply \( A^{-1} \) by \( B \):

\[
X = A^{-1} \cdot B = \begin{pmatrix} \frac{3}{2} & -2 \\ -\frac{1}{2} & 1 \end{pmatrix} \cdot \begin{pmatrix} 8 & 2 \\ 3 & 2 \end{pmatrix}.
\]

Let's calculate this matrix product:

\[
X_{11} = \left( \frac{3}{2} \times 8 \right) + \left( -2 \times 3 \right) = 12 - 6 = 6,
\]
\[
X_{12} = \left( \frac{3}{2} \times 2 \right) + \left( -2 \times 2 \right) = 3 - 4 = -1,
\]
\[
X_{21} = \left( -\frac{1}{2} \times 8 \right) + \left( 1 \times 3 \right) = -4 + 3 = -1,
\]
\[
X_{22} = \left( -\frac{1}{2} \times 2 \right) + \left( 1 \times 2 \right) = -1 + 2 = 1.
\]

Thus, the matrix \( X \) is:

\[
X = \begin{pmatrix} 6 & -1 \\ -1 & 1 \end{pmatrix}.
\]

### Final Answer:
\[
X = \begin{pmatrix} 6 & -1 \\ -1 & 1 \end{pmatrix}.
\]

Let me know if you would like further details or have questions! Here are some related questions to explore:

1. How do you compute the determinant of a 3x3 matrix?
2. Can all square matrices be inverted? Why or why not?
3. What is the geometric interpretation of the inverse of a matrix?
4. How is matrix multiplication used in solving systems of linear equations?
5. What are the properties of matrix inverses?

**Tip:** When calculating the inverse of a matrix, always check that the determinant is not zero. If it is, the matrix has no inverse (it's called *singular*).

Diketahui matriks-matriks A = (2 4; 1 3) dan B = (8 2; 3 2). Tentukan matriks X jika A . X = B.

This problem involves finding matrix X by solving the matrix equation A . X = B using matrix inversion and multiplication. Learn how to compute the inverse of matrix A and perform the matrix multiplication to solve for X.

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