If `A=({:(-2,2),(2,-2):})`, then which one of the following is correct ?

Options

A. `A^(2)=-2A`
B. `A^(2)=-4A`
C. `A^(2)=-3A`
D. `A^(2)=4A`

Explanation

To solve the problem, we need to find \( A^2 \) for the matrix \( A = \begin{pmatrix} -2 & 2 \\ 2 & -2 \end{pmatrix} \) and compare it with \( -4A \) to determine which option is correct.

### Step 1: Calculate \( A^2 \)

We start by calculating \( A^2 \) which is \( A \times A \).

\[ A^2 = \begin{pmatrix} -2 & 2 \\ 2 & -2 \end{pmatrix} \times \begin{pmatrix} -2 & 2 \\ 2 & -2 \end{pmatrix} \]

Using the matrix multiplication formula, we calculate each element of the resulting matrix:

- The element at position (1,1):
\[ (-2) \times (-2) + (2) \times (2) = 4 + 4 = 8 \]

- The element at position (1,2):
\[ (-2) \times (2) + (2) \times (-2) = -4 - 4 = -8 \]

- The element at position (2,1):
\[ (2) \times (-2) + (-2) \times (2) = -4 - 4 = -8 \]

- The element at position (2,2):
\[ (2) \times (2) + (-2) \times (-2) = 4 + 4 = 8 \]

Thus, we have:
\[ A^2 = \begin{pmatrix} 8 & -8 \\ -8 & 8 \end{pmatrix} \]

### Step 2: Calculate \( -4A \)

Next, we calculate \( -4A \):
\[ -4A = -4 \times \begin{pmatrix} -2 & 2 \\ 2 & -2 \end{pmatrix} = \begin{pmatrix} 8 & -8 \\ -8 & 8 \end{pmatrix} \]

### Step 3: Compare \( A^2 \) and \( -4A \)

Now we compare \( A^2 \) and \( -4A \):
\[ A^2 = \begin{pmatrix} 8 & -8 \\ -8 & 8 \end{pmatrix} \]
\[ -4A = \begin{pmatrix} 8 & -8 \\ -8 & 8 \end{pmatrix} \]

Since \( A^2 = -4A \), we conclude that this relationship holds true.

### Conclusion

Thus, the correct option is **B**, as \( A^2 = -4A \).

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