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The problem asks you to calculate \( A^T - 2B - 4C \), where the matrices \( A \), \( B \), and \( C \) are provided as:

\[
A = \begin{bmatrix} 3 & -4 \\ 2 & 1 \end{bmatrix}, \quad B = \begin{bmatrix} -3 & -2 \\ -1 & 5 \end{bmatrix}, \quad C = \begin{bmatrix} 5 & 4 \\ -2 & -1 \end{bmatrix}
\]

### Step 1: Find the transpose of matrix \( A \), denoted as \( A^T \).

\[
A^T = \begin{bmatrix} 3 & 2 \\ -4 & 1 \end{bmatrix}
\]

### Step 2: Multiply matrix \( B \) by 2.

\[
2B = 2 \times \begin{bmatrix} -3 & -2 \\ -1 & 5 \end{bmatrix} = \begin{bmatrix} -6 & -4 \\ -2 & 10 \end{bmatrix}
\]

### Step 3: Multiply matrix \( C \) by 4.

\[
4C = 4 \times \begin{bmatrix} 5 & 4 \\ -2 & -1 \end{bmatrix} = \begin{bmatrix} 20 & 16 \\ -8 & -4 \end{bmatrix}
\]

### Step 4: Subtract \( 2B \) and \( 4C \) from \( A^T \).

Now, calculate:

\[
A^T - 2B - 4C = \begin{bmatrix} 3 & 2 \\ -4 & 1 \end{bmatrix} - \begin{bmatrix} -6 & -4 \\ -2 & 10 \end{bmatrix} - \begin{bmatrix} 20 & 16 \\ -8 & -4 \end{bmatrix}
\]

Perform the subtraction step by step:

1. First, subtract \( 2B \) from \( A^T \):

\[
A^T - 2B = \begin{bmatrix} 3 & 2 \\ -4 & 1 \end{bmatrix} - \begin{bmatrix} -6 & -4 \\ -2 & 10 \end{bmatrix} = \begin{bmatrix} 3 + 6 & 2 + 4 \\ -4 + 2 & 1 - 10 \end{bmatrix} = \begin{bmatrix} 9 & 6 \\ -2 & -9 \end{bmatrix}
\]

2. Next, subtract \( 4C \):

\[
A^T - 2B - 4C = \begin{bmatrix} 9 & 6 \\ -2 & -9 \end{bmatrix} - \begin{bmatrix} 20 & 16 \\ -8 & -4 \end{bmatrix} = \begin{bmatrix} 9 - 20 & 6 - 16 \\ -2 - (-8) & -9 - (-4) \end{bmatrix} = \begin{bmatrix} -11 & -10 \\ 6 & -5 \end{bmatrix}
\]

### Final Answer:

\[
A^T - 2B - 4C = \begin{bmatrix} -11 & -10 \\ 6 & -5 \end{bmatrix}
\]

Let me know if you'd like more details, or if you have any questions!

### Related Questions:
1. How do you find the determinant of matrix \( A \)?
2. What is the inverse of matrix \( B \), if it exists?
3. How do you calculate the rank of matrix \( C \)?
4. Can you describe the geometric interpretation of matrix transposition?
5. What are eigenvalues and eigenvectors, and how would you compute them for matrix \( A \)?

### Tip:
When subtracting or adding matrices, make sure both matrices are of the same dimension and perform element-wise operations to avoid errors.

Diketahui A = [3, -4; 2, 1], B = [-3, -2; -1, 5] dan C = [5, 4; -2, -1]. Tentukan A^T - 2B - 4C!

Learn how to solve the matrix expression A^T - 2B - 4C where matrices A, B, and C are given. This problem covers matrix transpose, scalar multiplication, and matrix subtraction, with step-by-step explanations suitable for advanced high school or college-level students.

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