Let's solve the given matrix operation problem step by step.

The problem asks for the result of the following matrix operation:

$3 \begin{pmatrix} 2 \\ 0 \\ 2 \end{pmatrix} - \begin{pmatrix} -1 \\ -2 \\ 0 \end{pmatrix}$

Step 1: Scalar Multiplication

First, multiply the scalar $3$ with each element of the first matrix:

$3 \times \begin{pmatrix} 2 \\ 0 \\ 2 \end{pmatrix} = \begin{pmatrix} 6 \\ 0 \\ 6 \end{pmatrix}$

Step 2: Subtraction of Matrices

Now subtract the second matrix from the result of the scalar multiplication:

$\begin{pmatrix} 6 \\ 0 \\ 6 \end{pmatrix} - \begin{pmatrix} -1 \\ -2 \\ 0 \end{pmatrix} = \begin{pmatrix} 6 - (-1) \\ 0 - (-2) \\ 6 - 0 \end{pmatrix} = \begin{pmatrix} 6 + 1 \\ 0 + 2 \\ 6 \end{pmatrix} = \begin{pmatrix} 7 \\ 2 \\ 6 \end{pmatrix}$

Answer:

The result is the matrix:

$\begin{pmatrix} 7 \\ 2 \\ 6 \end{pmatrix}$

It looks like the correct answer corresponds to option A based on the choices.

Do you have any questions or want further details? Here are 5 related questions to expand this topic:

1. How do you perform matrix addition and subtraction in general?
2. Can you explain how to multiply matrices of different dimensions?
3. What happens when you multiply two matrices instead of using scalar multiplication?
4. What are some common applications of matrix operations in real-world problems?
5. How does scalar multiplication differ from matrix-matrix multiplication?

Tip: Always check that the dimensions of matrices are compatible before performing operations like addition, subtraction, or multiplication.