Let's break down the given matrix equation step by step:

The expression is:

$4 \left( \begin{bmatrix} 1 & -4 \\ -6 & 5 \\ -4 & 2 \end{bmatrix} + \begin{bmatrix} -4 & -6 \\ 1 & 3 \\ -1 & -4 \end{bmatrix} \right) 4$

Step 1: Add the two matrices inside the parentheses.

$\begin{bmatrix} 1 & -4 \\ -6 & 5 \\ -4 & 2 \end{bmatrix} + \begin{bmatrix} -4 & -6 \\ 1 & 3 \\ -1 & -4 \end{bmatrix} = \begin{bmatrix} 1 + (-4) & -4 + (-6) \\ -6 + 1 & 5 + 3 \\ -4 + (-1) & 2 + (-4) \end{bmatrix}$

This results in:

$\begin{bmatrix} -3 & -10 \\ -5 & 8 \\ -5 & -2 \end{bmatrix}$

Step 2: Multiply the result by 4.

We need to multiply each element of the resulting matrix by 4:

$4 \times \begin{bmatrix} -3 & -10 \\ -5 & 8 \\ -5 & -2 \end{bmatrix} = \begin{bmatrix} 4 \times (-3) & 4 \times (-10) \\ 4 \times (-5) & 4 \times 8 \\ 4 \times (-5) & 4 \times (-2) \end{bmatrix}$

This gives:

$\begin{bmatrix} -12 & -40 \\ -20 & 32 \\ -20 & -8 \end{bmatrix}$

Final Answer:

The result of the given matrix operation is:

$\begin{bmatrix} -12 & -40 \\ -20 & 32 \\ -20 & -8 \end{bmatrix}$

Let me know if you'd like more details on any specific step!

Further Questions:

1. How do you perform matrix addition in general?
2. What happens when we multiply a matrix by a scalar?
3. Can you perform matrix multiplication between two matrices? What's the condition?
4. What is the difference between scalar and matrix multiplication?
5. Can a non-square matrix be multiplied by itself?

Tip: Always verify matrix dimensions before performing addition or multiplication to ensure operations are valid!