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We are given three matrices \(A\), \(B\), and \(C\), which satisfy the equation:

\[
C = \frac{1}{2} AB
\]

with matrix \(A\) and \(B\) given as:

\[
A = \begin{pmatrix} 0 & 1 \\ 2 & 2 \end{pmatrix}, \quad B = \begin{pmatrix} 3 & -1 \\ 0 & 6 \end{pmatrix}.
\]

We need to find the inverse of matrix \(C\).

### Step 1: Compute the product \(AB\)
First, we calculate the product \(A \cdot B\):

\[
A = \begin{pmatrix} 0 & 1 \\ 2 & 2 \end{pmatrix}, \quad B = \begin{pmatrix} 3 & -1 \\ 0 & 6 \end{pmatrix}
\]

\[
AB = \begin{pmatrix} 0 \cdot 3 + 1 \cdot 0 & 0 \cdot (-1) + 1 \cdot 6 \\ 2 \cdot 3 + 2 \cdot 0 & 2 \cdot (-1) + 2 \cdot 6 \end{pmatrix}
= \begin{pmatrix} 0 & 6 \\ 6 & 10 \end{pmatrix}
\]

### Step 2: Multiply by \(\frac{1}{2}\)
Now, multiply \(AB\) by \(\frac{1}{2}\):

\[
C = \frac{1}{2} \cdot \begin{pmatrix} 0 & 6 \\ 6 & 10 \end{pmatrix} = \begin{pmatrix} 0 & 3 \\ 3 & 5 \end{pmatrix}
\]

### Step 3: Find the inverse of matrix \(C\)
We now need to find the inverse of matrix \(C = \begin{pmatrix} 0 & 3 \\ 3 & 5 \end{pmatrix}\). The inverse of a 2x2 matrix \(\begin{pmatrix} a & b \\ c & d \end{pmatrix}\) is given by:

\[
C^{-1} = \frac{1}{ad - bc} \begin{pmatrix} d & -b \\ -c & a \end{pmatrix}
\]

For \(C = \begin{pmatrix} 0 & 3 \\ 3 & 5 \end{pmatrix}\), we have:

\[
ad - bc = (0)(5) - (3)(3) = -9
\]

Thus, the inverse is:

\[
C^{-1} = \frac{1}{-9} \begin{pmatrix} 5 & -3 \\ -3 & 0 \end{pmatrix} = \begin{pmatrix} -\frac{5}{9} & \frac{1}{3} \\ \frac{1}{3} & 0 \end{pmatrix}
\]

Thus, the inverse of matrix \(C\) is:

\[
C^{-1} = \begin{pmatrix} -\frac{5}{9} & \frac{1}{3} \\ \frac{1}{3} & 0 \end{pmatrix}
\]

---

Would you like further details or have any questions? Here are some additional questions for exploration:

1. How do you calculate the inverse of a matrix with more than two rows and columns?
2. Why is the determinant important in finding the inverse of a matrix?
3. What are the conditions for a matrix to have an inverse?
4. How do you find the product of two matrices step-by-step?
5. What is the geometric interpretation of a matrix inverse?

**Tip:** Always check the determinant of a matrix before attempting to find its inverse. If the determinant is zero, the matrix is not invertible.

Matriks A, B, dan C memenuhi persamaan C = 1/2 AB. Jika A = [[0,1],[2,2]] dan B = [[3,-1],[0,6]], tentukan invers dari matriks C.

Learn how to find the inverse of matrix C where C = 1/2 AB, given matrices A = [[0,1],[2,2]] and B = [[3,-1],[0,6]]. Step-by-step solution includes matrix multiplication, scaling, and calculating the inverse of a 2x2 matrix.

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