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The problem given asks to perform the following matrix operation:

\[
2(A - B) + 4(C - A) + 2(A - C) - 2C + 3B
\]

where the matrices are defined as:
\[
A = \begin{bmatrix} 0 & 3 & 4 \\ 4 & -1 & 2 \end{bmatrix}, \quad B = \begin{bmatrix} 7 & 9 & -1 \\ 6 & -2 & 3 \end{bmatrix}, \quad C = \begin{bmatrix} 12 & 6 & 4 \\ 9 & 1 & 0 \end{bmatrix}
\]

We will break this down step-by-step.

### Step 1: Compute \(A - B\)
\[
A - B = \begin{bmatrix} 0 & 3 & 4 \\ 4 & -1 & 2 \end{bmatrix} - \begin{bmatrix} 7 & 9 & -1 \\ 6 & -2 & 3 \end{bmatrix} = \begin{bmatrix} 0 - 7 & 3 - 9 & 4 + 1 \\ 4 - 6 & -1 + 2 & 2 - 3 \end{bmatrix}
= \begin{bmatrix} -7 & -6 & 5 \\ -2 & 1 & -1 \end{bmatrix}
\]

### Step 2: Compute \(C - A\)
\[
C - A = \begin{bmatrix} 12 & 6 & 4 \\ 9 & 1 & 0 \end{bmatrix} - \begin{bmatrix} 0 & 3 & 4 \\ 4 & -1 & 2 \end{bmatrix} = \begin{bmatrix} 12 - 0 & 6 - 3 & 4 - 4 \\ 9 - 4 & 1 + 1 & 0 - 2 \end{bmatrix}
= \begin{bmatrix} 12 & 3 & 0 \\ 5 & 2 & -2 \end{bmatrix}
\]

### Step 3: Compute \(A - C\)
\[
A - C = \begin{bmatrix} 0 & 3 & 4 \\ 4 & -1 & 2 \end{bmatrix} - \begin{bmatrix} 12 & 6 & 4 \\ 9 & 1 & 0 \end{bmatrix} = \begin{bmatrix} 0 - 12 & 3 - 6 & 4 - 4 \\ 4 - 9 & -1 - 1 & 2 - 0 \end{bmatrix}
= \begin{bmatrix} -12 & -3 & 0 \\ -5 & -2 & 2 \end{bmatrix}
\]

### Step 4: Multiply and sum terms
Now we apply the multiplications and sums from the original expression:

1. \( 2(A - B) = 2 \times \begin{bmatrix} -7 & -6 & 5 \\ -2 & 1 & -1 \end{bmatrix} = \begin{bmatrix} -14 & -12 & 10 \\ -4 & 2 & -2 \end{bmatrix} \)
   
2. \( 4(C - A) = 4 \times \begin{bmatrix} 12 & 3 & 0 \\ 5 & 2 & -2 \end{bmatrix} = \begin{bmatrix} 48 & 12 & 0 \\ 20 & 8 & -8 \end{bmatrix} \)
   
3. \( 2(A - C) = 2 \times \begin{bmatrix} -12 & -3 & 0 \\ -5 & -2 & 2 \end{bmatrix} = \begin{bmatrix} -24 & -6 & 0 \\ -10 & -4 & 4 \end{bmatrix} \)
   
4. \( -2C = -2 \times \begin{bmatrix} 12 & 6 & 4 \\ 9 & 1 & 0 \end{bmatrix} = \begin{bmatrix} -24 & -12 & -8 \\ -18 & -2 & 0 \end{bmatrix} \)
   
5. \( 3B = 3 \times \begin{bmatrix} 7 & 9 & -1 \\ 6 & -2 & 3 \end{bmatrix} = \begin{bmatrix} 21 & 27 & -3 \\ 18 & -6 & 9 \end{bmatrix} \)

### Step 5: Combine all terms
Now we add all the resulting matrices:

\[
2(A - B) + 4(C - A) + 2(A - C) - 2C + 3B = \begin{bmatrix} -14 & -12 & 10 \\ -4 & 2 & -2 \end{bmatrix} + \begin{bmatrix} 48 & 12 & 0 \\ 20 & 8 & -8 \end{bmatrix}
+ \begin{bmatrix} -24 & -6 & 0 \\ -10 & -4 & 4 \end{bmatrix} + \begin{bmatrix} -24 & -12 & -8 \\ -18 & -2 & 0 \end{bmatrix} + \begin{bmatrix} 21 & 27 & -3 \\ 18 & -6 & 9 \end{bmatrix}
\]

Now adding the elements together:

\[
= \begin{bmatrix} (-14 + 48 - 24 - 24 + 21) & (-12 + 12 - 6 - 12 + 27) & (10 + 0 + 0 - 8 - 3) \\ (-4 + 20 - 10 - 18 + 18) & (2 + 8 - 4 - 2 - 6) & (-2 - 8 + 4 + 0 + 9) \end{bmatrix}
\]

\[
= \begin{bmatrix} 7 & 9 & -1 \\ 6 & -2 & 3 \end{bmatrix}
\]

Thus, the final result of the operation is:

\[
\boxed{\begin{bmatrix} 7 & 9 & -1 \\ 6 & -2 & 3 \end{bmatrix}}
\]

This matches the first answer option.

Would you like further details or clarifications? 

### Follow-up Questions:
1. What are the conditions for matrix addition or subtraction to be valid?
2. How do you determine if two matrices can be multiplied?
3. What are the steps for multiplying a matrix by a scalar?
4. How can you verify the correctness of matrix operations?
5. What are some common properties of matrix operations (e.g., distributive property)?

### Tip:
Always ensure the dimensions of the matrices match for addition or subtraction operations.

Given A = [[0, 3, 4], [4, -1, 2]], B = [[7, 9, -1], [6, -2, 3]], C = [[12, 6, 4], [9, 1, 0]], perform the operation: 2(A - B) + 4(C - A) + 2(A - C) - 2C + 3B.

Perform matrix operations involving addition, subtraction, and scalar multiplication. Given matrices A, B, and C, solve the expression 2(A - B) + 4(C - A) + 2(A - C) - 2C + 3B step-by-step. Ideal for university students studying linear algebra.

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