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The problem asks to evaluate the expression \( A(B - C) \), where:

- \( A = \begin{bmatrix} 0 & -2 & 0 & 1 \\ 11 & 0 & 1 & 0 \end{bmatrix} \),
- \( B = \begin{bmatrix} 0 & -2 \\ 1 & 1 \\ -1 & 3 \\ 7 & 0 \end{bmatrix} \),
- \( C = \begin{bmatrix} 1 & -1 \\ 1 & 1 \\ 1 & 1 \\ 1 & 1 \end{bmatrix} \).

### Step 1: Subtract Matrices \( B \) and \( C \)

We need to calculate \( B - C \):
\[
B - C = \begin{bmatrix} 0 & -2 \\ 1 & 1 \\ -1 & 3 \\ 7 & 0 \end{bmatrix} - \begin{bmatrix} 1 & -1 \\ 1 & 1 \\ 1 & 1 \\ 1 & 1 \end{bmatrix} = \begin{bmatrix} 0-1 & -2-(-1) \\ 1-1 & 1-1 \\ -1-1 & 3-1 \\ 7-1 & 0-1 \end{bmatrix}
\]
\[
= \begin{bmatrix} -1 & -1 \\ 0 & 0 \\ -2 & 2 \\ 6 & -1 \end{bmatrix}
\]

### Step 2: Multiply \( A \) by \( (B - C) \)

Now, we need to compute \( A(B - C) \):
\[
A = \begin{bmatrix} 0 & -2 & 0 & 1 \\ 11 & 0 & 1 & 0 \end{bmatrix}, \quad (B - C) = \begin{bmatrix} -1 & -1 \\ 0 & 0 \\ -2 & 2 \\ 6 & -1 \end{bmatrix}
\]
Multiply the matrices:
\[
A(B - C) = \begin{bmatrix} 0 & -2 & 0 & 1 \\ 11 & 0 & 1 & 0 \end{bmatrix} \times \begin{bmatrix} -1 & -1 \\ 0 & 0 \\ -2 & 2 \\ 6 & -1 \end{bmatrix}
\]

#### First row of \( A(B - C) \):
\[
[0, -2, 0, 1] \times \begin{bmatrix} -1 \\ 0 \\ -2 \\ 6 \end{bmatrix} = 0(-1) + (-2)(0) + 0(-2) + 1(6) = 6
\]
\[
[0, -2, 0, 1] \times \begin{bmatrix} -1 \\ 0 \\ 2 \\ -1 \end{bmatrix} = 0(-1) + (-2)(0) + 0(2) + 1(-1) = -1
\]

#### Second row of \( A(B - C) \):
\[
[11, 0, 1, 0] \times \begin{bmatrix} -1 \\ 0 \\ -2 \\ 6 \end{bmatrix} = 11(-1) + 0(0) + 1(-2) + 0(6) = -11 - 2 = -13
\]
\[
[11, 0, 1, 0] \times \begin{bmatrix} -1 \\ 0 \\ 2 \\ -1 \end{bmatrix} = 11(-1) + 0(0) + 1(2) + 0(-1) = -11 + 2 = -9
\]

### Final result:
\[
A(B - C) = \begin{bmatrix} 6 & -1 \\ -13 & -9 \end{bmatrix}
\]

Let me know if you want any clarifications or further details. Here are some related questions:

1. How can we verify if two matrices are conformable for multiplication?
2. What are some properties of matrix multiplication?
3. How does the distributive property work with matrix operations?
4. In what scenarios would matrix subtraction be undefined?
5. What are the real-world applications of multiplying matrices?

**Tip:** Always ensure that matrix dimensions are compatible before performing operations like multiplication or subtraction.

Evaluate A(B - C) where A = [[0, -2, 0, 1], [11, 0, 1, 0]], B = [[0, -2], [1, 1], [-1, 3], [7, 0]], C = [[1, -1], [1, 1], [1, 1], [1, 1]].

Learn how to evaluate A(B - C) with a detailed solution. This problem involves matrix subtraction and matrix multiplication, including how to ensure matrix dimensions are compatible for operations. Suitable for college-level linear algebra students.

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