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):