If `A=[(1,-1),(-1,1)], "then" " "A^(3)`

Options

A. 3A
B. `-3A`
C. 4A
D. `-4A`

Answer

To find \( A^3 \) for the matrix \( A = \begin{pmatrix} 1 & -1 \\ -1 & 1 \end{pmatrix} \), we will follow these steps:

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

To find \( A^2 \), we multiply \( A \) by itself:

\[ A^2 = A \cdot A = \begin{pmatrix} 1 & -1 \\ -1 & 1 \end{pmatrix} \cdot \begin{pmatrix} 1 & -1 \\ -1 & 1 \end{pmatrix} \]

Calculating the elements of the resulting matrix:

- First row, first column:
\[ 1 \cdot 1 + (-1) \cdot (-1) = 1 + 1 = 2 \]

- First row, second column:
\[ 1 \cdot (-1) + (-1) \cdot 1 = -1 - 1 = -2 \]

- Second row, first column:
\[ (-1) \cdot 1 + 1 \cdot (-1) = -1 - 1 = -2 \]

- Second row, second column:
\[ (-1) \cdot (-1) + 1 \cdot 1 = 1 + 1 = 2 \]

Thus, we have:

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

### Step 2: Calculate \( A^3 \)

Now, we need to calculate \( A^3 \) by multiplying \( A^2 \) by \( A \):

\[ A^3 = A^2 \cdot A = \begin{pmatrix} 2 & -2 \\ -2 & 2 \end{pmatrix} \cdot \begin{pmatrix} 1 & -1 \\ -1 & 1 \end{pmatrix} \]

Calculating the elements of the resulting matrix:

- First row, first column:
\[ 2 \cdot 1 + (-2) \cdot (-1) = 2 + 2 = 4 \]

- First row, second column:
\[ 2 \cdot (-1) + (-2) \cdot 1 = -2 - 2 = -4 \]

- Second row, first column:
\[ (-2) \cdot 1 + 2 \cdot (-1) = -2 - 2 = -4 \]

- Second row, second column:
\[ (-2) \cdot (-1) + 2 \cdot 1 = 2 + 2 = 4 \]

Thus, we have:

\[ A^3 = \begin{pmatrix} 4 & -4 \\ -4 & 4 \end{pmatrix} \]

### Final Result

The final result for \( A^3 \) is:

\[ A^3 = \begin{pmatrix} 4 & -4 \\ -4 & 4 \end{pmatrix} \]