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The problem gives the following information:

Matrices:
\[
A = \begin{pmatrix} 1 & 2 \\ 1 & 3 \end{pmatrix}, \quad B = \begin{pmatrix} 3 & 1 \\ 2 & 4 \end{pmatrix}
\]
It is also stated that \( AC = B \), and we need to determine the determinant of matrix \( C \).

### Step 1: Set up the equation \( AC = B \)
Let \( C = \begin{pmatrix} c_{11} & c_{12} \\ c_{21} & c_{22} \end{pmatrix} \). From the equation \( AC = B \), we have:

\[
\begin{pmatrix} 1 & 2 \\ 1 & 3 \end{pmatrix} \begin{pmatrix} c_{11} & c_{12} \\ c_{21} & c_{22} \end{pmatrix} = \begin{pmatrix} 3 & 1 \\ 2 & 4 \end{pmatrix}
\]

### Step 2: Multiply matrices on the left side
Performing the matrix multiplication for \( A \cdot C \):

\[
AC = \begin{pmatrix} 1 \cdot c_{11} + 2 \cdot c_{21} & 1 \cdot c_{12} + 2 \cdot c_{22} \\ 1 \cdot c_{11} + 3 \cdot c_{21} & 1 \cdot c_{12} + 3 \cdot c_{22} \end{pmatrix}
\]

This results in the system of equations:
1. \( c_{11} + 2c_{21} = 3 \)
2. \( c_{12} + 2c_{22} = 1 \)
3. \( c_{11} + 3c_{21} = 2 \)
4. \( c_{12} + 3c_{22} = 4 \)

### Step 3: Solve for elements of \( C \)
From equations (1) and (3):

\[
\begin{aligned}
c_{11} + 2c_{21} &= 3 \\
c_{11} + 3c_{21} &= 2
\end{aligned}
\]

Subtract the first equation from the second:
\[
( c_{11} + 3c_{21} ) - ( c_{11} + 2c_{21} ) = 2 - 3 \implies c_{21} = -1
\]

Substitute \( c_{21} = -1 \) into the first equation:
\[
c_{11} + 2(-1) = 3 \implies c_{11} - 2 = 3 \implies c_{11} = 5
\]

Now, from equations (2) and (4):

\[
\begin{aligned}
c_{12} + 2c_{22} &= 1 \\
c_{12} + 3c_{22} &= 4
\end{aligned}
\]

Subtract the first equation from the second:
\[
( c_{12} + 3c_{22} ) - ( c_{12} + 2c_{22} ) = 4 - 1 \implies c_{22} = 3
\]

Substitute \( c_{22} = 3 \) into the second equation:
\[
c_{12} + 2(3) = 1 \implies c_{12} + 6 = 1 \implies c_{12} = -5
\]

Thus, the matrix \( C \) is:
\[
C = \begin{pmatrix} 5 & -5 \\ -1 & 3 \end{pmatrix}
\]

### Step 4: Compute the determinant of \( C \)
The determinant of \( C \) is:
\[
\det(C) = (5)(3) - (-5)(-1) = 15 - 5 = 10
\]

### Final Answer:
The determinant of matrix \( C \) is \( \boxed{10} \).

---

Would you like any further clarifications or details on this process? Here are some related questions you might find useful:

1. What is the general process of matrix multiplication?
2. How do you solve systems of linear equations using substitution?
3. What are the properties of determinants for 2x2 matrices?
4. Can you explain the relationship between matrices and linear transformations?
5. How is the inverse of a matrix computed?

**Tip:** When solving systems of equations, organizing your work and checking solutions step-by-step can help prevent errors.

Diketahui matriks A = [[1, 2], [1, 3]], B = [[3, 1], [2, 4]], dan memenuhi AC = B. Determinan matriks C adalah ...

Learn how to calculate the determinant of matrix C, given matrices A and B, with the condition AC = B. This problem covers key concepts in matrix multiplication, solving systems of equations, and determinant calculations. Suitable for high school students in Grades 10-12.

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