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To find the inverses of the given matrices \( A \), \( B \), and \( C \), we need to check if each matrix is invertible. A matrix is invertible if its determinant is non-zero. If the determinant is zero, the matrix is singular, and no inverse exists.

### Matrix \( A \):
\[
A = \begin{pmatrix} 3 & 1 \\ 3 & 2 \end{pmatrix}
\]
The determinant of \( A \) is:
\[
\text{det}(A) = (3)(2) - (1)(3) = 6 - 3 = 3
\]
Since the determinant of \( A \) is 3 (non-zero), the matrix is invertible. The inverse of a 2x2 matrix \( \begin{pmatrix} a & b \\ c & d \end{pmatrix} \) is given by:
\[
A^{-1} = \frac{1}{\text{det}(A)} \begin{pmatrix} d & -b \\ -c & a \end{pmatrix}
\]
Applying this formula to matrix \( A \):
\[
A^{-1} = \frac{1}{3} \begin{pmatrix} 2 & -1 \\ -3 & 3 \end{pmatrix} = \begin{pmatrix} \frac{2}{3} & -\frac{1}{3} \\ -1 & 1 \end{pmatrix}
\]

### Matrix \( B \):
\[
B = \begin{pmatrix} 1 & 4 \\ -3 & 2 \end{pmatrix}
\]
The determinant of \( B \) is:
\[
\text{det}(B) = (1)(2) - (4)(-3) = 2 + 12 = 14
\]
Since the determinant of \( B \) is 14 (non-zero), the matrix is invertible. Using the inverse formula:
\[
B^{-1} = \frac{1}{14} \begin{pmatrix} 2 & -4 \\ 3 & 1 \end{pmatrix} = \begin{pmatrix} \frac{2}{14} & -\frac{4}{14} \\ \frac{3}{14} & \frac{1}{14} \end{pmatrix} = \begin{pmatrix} \frac{1}{7} & -\frac{2}{7} \\ \frac{3}{14} & \frac{1}{14} \end{pmatrix}
\]

### Matrix \( C \):
\[
C = \begin{pmatrix} 3 & 4 \\ -3 & -6 \end{pmatrix}
\]
The determinant of \( C \) is:
\[
\text{det}(C) = (3)(-6) - (4)(-3) = -18 + 12 = -6
\]
Since the determinant of \( C \) is -6 (non-zero), the matrix is invertible. Using the inverse formula:
\[
C^{-1} = \frac{1}{-6} \begin{pmatrix} -6 & -4 \\ 3 & 3 \end{pmatrix} = \begin{pmatrix} 1 & \frac{2}{3} \\ -\frac{1}{2} & -\frac{1}{2} \end{pmatrix}
\]

### Summary:
- \( A^{-1} = \begin{pmatrix} \frac{2}{3} & -\frac{1}{3} \\ -1 & 1 \end{pmatrix} \)
- \( B^{-1} = \begin{pmatrix} \frac{1}{7} & -\frac{2}{7} \\ \frac{3}{14} & \frac{1}{14} \end{pmatrix} \)
- \( C^{-1} = \begin{pmatrix} 1 & \frac{2}{3} \\ -\frac{1}{2} & -\frac{1}{2} \end{pmatrix} \)

Would you like more details on the calculation steps or have any other questions?

Here are some follow-up questions for further exploration:
1. What is the significance of the determinant in determining the invertibility of a matrix?
2. How does matrix inversion apply to solving systems of linear equations?
3. Can we use a different method, like Gaussian elimination, to find the inverse of a matrix?
4. How does the inverse of a matrix relate to its transpose and adjugate?
5. What is the computational complexity of finding the inverse of a matrix in larger systems?

**Tip:** Always check the determinant before attempting to find the inverse of a matrix, as it will save time if the matrix is singular.

Find where possible the inverses of the following matrices:
A = (3 1 | 3 2), B = (1 4 | -3 2), C = (3 4 | -3 -6).
If an inverse cannot be obtained, state the reason behind this result.

This solution explains how to find the inverse of the matrices A, B, and C using the determinant method. Detailed steps include checking if each matrix is invertible by calculating the determinant, followed by the formula for finding the inverse of a 2x2 matrix.

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