Question

Let \( A = \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix} \). Then, \( A^5 = \):

• A. \( \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix} \)
• B. \( \begin{bmatrix} 0 & 1 \\ 0 & 1 \end{bmatrix} \)
• C. \( \begin{bmatrix} 0 & 5 \\ 5 & 0 \end{bmatrix} \)
• D. \( \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} \)

Hint:

For powers of matrices, look for patterns such as repetition or identity matrix results. If \( A^2 = I \), then \( A^n \) cycles every 2 terms.

Answer 1

The Correct Option is A

Let \( A = \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix} \). Note that: \[ A^2 = \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix} \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix} = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} = I \] Since \( A^2 = I \), then powers of \( A \) cycle every 2 steps: \[ A^3 = A^2 \cdot A = I \cdot A = A, \quad A^4 = A^2 \cdot A^2 = I \cdot I = I, \quad A^5 = A^4 \cdot A = I \cdot A = A \] \[ \therefore A^5 = \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix} \]