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The problem in the image presents matrices \( A \), \( B \), and \( C \) and asks for the value of the expression \( A B + 2C \). Here's the data from the image:

- \( A = \begin{pmatrix} 4 & -2 \\ -4 & 1 \end{pmatrix} \)
- \( B = \begin{pmatrix} -3 & -2 \\ 1 & 5 \end{pmatrix} \)
- \( C = \begin{pmatrix} 4 & -3 \\ -6 & 0 \end{pmatrix} \)

We need to calculate the following:
\[
A B + 2C
\]

### Step-by-step solution:

1. **Matrix Multiplication \( A \times B \):**
   \[
   A B = \begin{pmatrix} 4 & -2 \\ -4 & 1 \end{pmatrix} \times \begin{pmatrix} -3 & -2 \\ 1 & 5 \end{pmatrix}
   \]
   We compute this by calculating each element of the resulting matrix.

2. **Multiplying matrix \( C \) by 2:**
   \[
   2C = 2 \times \begin{pmatrix} 4 & -3 \\ -6 & 0 \end{pmatrix} = \begin{pmatrix} 8 & -6 \\ -12 & 0 \end{pmatrix}
   \]

3. **Summing \( A B \) and \( 2C \):**
   Add the resulting matrices element-wise.

I'll now perform the calculations for \( A B \) and the final result.Here are the detailed steps and results:

1. **Matrix multiplication \( A \times B \):**
   \[
   A B = \begin{pmatrix} -14 & -18 \\ 13 & 13 \end{pmatrix}
   \]

2. **Multiplying matrix \( C \) by 2:**
   \[
   2C = \begin{pmatrix} 8 & -6 \\ -12 & 0 \end{pmatrix}
   \]

3. **Summing \( A B \) and \( 2C \):**
   \[
   A B + 2C = \begin{pmatrix} -6 & -24 \\ 1 & 13 \end{pmatrix}
   \]

Thus, the final result of \( A B + 2C \) is:
\[
\begin{pmatrix} -6 & -24 \\ 1 & 13 \end{pmatrix}
\]

Let me know if you'd like further clarifications or additional details!

Here are 5 related questions for further practice:
1. How would you calculate the determinant of matrix \( A \)?
2. What is the inverse of matrix \( B \), if it exists?
3. How do you perform matrix multiplication when one matrix has different dimensions?
4. What would be the result of \( A + B \) (element-wise addition)?
5. What is the trace of matrix \( C \)?

**Tip:** When performing matrix multiplication, remember that the number of columns in the first matrix must equal the number of rows in the second matrix.

Diketahui matriks A = [[4, -4], [-2, 1]], B = [[-3, 1], [-2, 5]], dan C = [[4, -6], [-3, 0]]. Nilai: A*B + 2C = ...

This problem involves calculating the result of the matrix operation A*B + 2C, using matrices A, B, and C. The solution includes step-by-step matrix multiplication, scalar multiplication, and addition. Suitable for advanced high school or university level students studying linear algebra.

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