If `A=[(0,1),(1,0)]` then `A^(4)=`

Options

A. `[(1,0),(0,1)]`
B. `[(1,1),(0,0)]`
C. `[(0,0),(1,1)]`
D. `[(0,1),(1,0)]`

Answer

To find \( A^4 \) for the matrix \( A = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} \), we will first calculate \( A^2 \) and then use that result to find \( A^4 \).

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

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

\[ A^2 = A \times A = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} \times \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} \]

Using matrix multiplication:

\[ A^2 = \begin{pmatrix} (0 \cdot 0 + 1 \cdot 1) & (0 \cdot 1 + 1 \cdot 0) \\ (1 \cdot 0 + 0 \cdot 1) & (1 \cdot 1 + 0 \cdot 0) \end{pmatrix} \]

Calculating each element:

- First row, first column: \( 0 \cdot 0 + 1 \cdot 1 = 1 \)
- First row, second column: \( 0 \cdot 1 + 1 \cdot 0 = 0 \)
- Second row, first column: \( 1 \cdot 0 + 0 \cdot 1 = 0 \)
- Second row, second column: \( 1 \cdot 1 + 0 \cdot 0 = 1 \)

Thus, we have:

\[ A^2 = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} \]

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

Now, we can find \( A^4 \) by multiplying \( A^2 \) by itself:

\[ A^4 = A^2 \times A^2 = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} \times \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} \]

Using matrix multiplication:

\[ A^4 = \begin{pmatrix} (1 \cdot 1 + 0 \cdot 0) & (1 \cdot 0 + 0 \cdot 1) \\ (0 \cdot 1 + 1 \cdot 0) & (0 \cdot 0 + 1 \cdot 1) \end{pmatrix} \]

Calculating each element:

- First row, first column: \( 1 \cdot 1 + 0 \cdot 0 = 1 \)
- First row, second column: \( 1 \cdot 0 + 0 \cdot 1 = 0 \)
- Second row, first column: \( 0 \cdot 1 + 1 \cdot 0 = 0 \)
- Second row, second column: \( 0 \cdot 0 + 1 \cdot 1 = 1 \)

Thus, we have:

\[ A^4 = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} \]

### Final Answer

The final result is:

\[ A^4 = I = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} \]