To find the inverse of a 2x2 matrix $C$, we can use the following formula for the inverse of a matrix $A = \begin{pmatrix} a & b \\ c & d \end{pmatrix}$:

$A^{-1} = \frac{1}{ad - bc} \begin{pmatrix} d & -b \\ -c & a \end{pmatrix}$

In this case, the matrix $C$ is given as:

$C = \begin{pmatrix} -2 & 3 \\ -2 & 8 \end{pmatrix}$

Let's apply the formula step by step:

1. Compute the determinant:
   The determinant of a 2x2 matrix is calculated as:

   $\text{det}(C) = (-2)(8) - (3)(-2) = -16 + 6 = -10$

2. Use the inverse formula:
   Now, use the formula for the inverse:

   $C^{-1} = \frac{1}{-10} \begin{pmatrix} 8 & -3 \\ 2 & -2 \end{pmatrix}$

   Multiplying by $\frac{1}{-10}$:

   $C^{-1} = \begin{pmatrix} \frac{8}{-10} & \frac{-3}{-10} \\ \frac{2}{-10} & \frac{-2}{-10} \end{pmatrix}$

3. Simplify the fractions:
   Simplifying the fractions:

   $C^{-1} = \begin{pmatrix} \frac{-4}{5} & \frac{3}{10} \\ \frac{1}{5} & \frac{-1}{5} \end{pmatrix}$

So, the inverse matrix is:

$C^{-1} = \begin{pmatrix} \frac{-4}{5} & \frac{3}{10} \\ \frac{1}{5} & \frac{-1}{5} \end{pmatrix}$

Would you like more details or have any questions about the solution?

Here are some related questions for practice:

1. How do you compute the inverse of a 3x3 matrix?
2. What happens if the determinant of a matrix is zero?
3. How can matrix inverses be used to solve systems of equations?
4. What is the relationship between the determinant and the invertibility of a matrix?
5. How do you interpret the inverse of a transformation matrix in geometry?

Tip: The inverse of a matrix exists only if the determinant is non-zero!