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To solve the expression \((A - B)^2\) using the given matrices \(A\) and \(B\), we can follow these steps:

### 1. Matrix Subtraction: \(A - B\)
Given matrices \(A\) and \(B\):
\[
A = \begin{pmatrix} 
0 & -1 & 2 \\ 
-1 & 1 & 1 \\ 
3 & 0 & 4 
\end{pmatrix},
\quad
B = \begin{pmatrix} 
4 & -1 & 2 \\ 
3 & 0 & 2 \\ 
0 & -1 & 4 
\end{pmatrix}
\]
First, we calculate \(A - B\):
\[
A - B = \begin{pmatrix} 
0-4 & -1-(-1) & 2-2 \\ 
-1-3 & 1-0 & 1-2 \\ 
3-0 & 0-(-1) & 4-4 
\end{pmatrix} 
= \begin{pmatrix} 
-4 & 0 & 0 \\ 
-4 & 1 & -1 \\ 
3 & 1 & 0 
\end{pmatrix}
\]

### 2. Squaring the Resulting Matrix: \((A - B)^2\)
Now, square the resulting matrix \(A - B\):
\[
(A - B)^2 = (A - B) \times (A - B)
\]
\[
= \begin{pmatrix} 
-4 & 0 & 0 \\ 
-4 & 1 & -1 \\ 
3 & 1 & 0 
\end{pmatrix}
\times
\begin{pmatrix} 
-4 & 0 & 0 \\ 
-4 & 1 & -1 \\ 
3 & 1 & 0 
\end{pmatrix}
\]
To perform matrix multiplication, each element of the resulting matrix is calculated as follows:

\[
\text{First row, first column: } (-4)(-4) + (0)(-4) + (0)(3) = 16 + 0 + 0 = 16
\]
\[
\text{First row, second column: } (-4)(0) + (0)(1) + (0)(1) = 0 + 0 + 0 = 0
\]
\[
\text{First row, third column: } (-4)(0) + (0)(-1) + (0)(0) = 0 + 0 + 0 = 0
\]
\[
\text{Second row, first column: } (-4)(-4) + (1)(-4) + (-1)(3) = 16 - 4 - 3 = 9
\]
\[
\text{Second row, second column: } (-4)(0) + (1)(1) + (-1)(1) = 0 + 1 - 1 = 0
\]
\[
\text{Second row, third column: } (-4)(0) + (1)(-1) + (-1)(0) = 0 - 1 + 0 = -1
\]
\[
\text{Third row, first column: } (3)(-4) + (1)(-4) + (0)(3) = -12 - 4 + 0 = -16
\]
\[
\text{Third row, second column: } (3)(0) + (1)(1) + (0)(1) = 0 + 1 + 0 = 1
\]
\[
\text{Third row, third column: } (3)(0) + (1)(-1) + (0)(0) = 0 - 1 + 0 = -1
\]

Thus, the resulting matrix is:
\[
(A - B)^2 = \begin{pmatrix} 
16 & 0 & 0 \\ 
9 & 0 & -1 \\ 
-16 & 1 & -1 
\end{pmatrix}
\]

Would you like any additional details or have further questions?

### Related Questions:
1. What is the determinant of the matrix \((A - B)^2\)?
2. How do you find the inverse of a matrix, and can \((A - B)^2\) be inverted?
3. How does matrix subtraction change when one matrix is a scalar multiple of the other?
4. What is the trace of the matrix \((A - B)\)?
5. How does the result of \((A + B)^2\) compare to \((A - B)^2\)?

### Tip:
When squaring a matrix, remember that the operation is not the same as element-wise squaring. Matrix multiplication involves summing the products of corresponding elements in the rows and columns.

Learn how to solve the matrix expression (A - B)^2 step-by-step. This problem involves matrix subtraction and multiplication, with detailed calculations provided. Suitable for advanced high school students and beyond.

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