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.