Question

If A = \(\begin{pmatrix} 0 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 0 & 0 \\ \end{pmatrix}\), then A⁴ is equal to

• A. A
• B. 2A
• C. I
• D. 4A

Answer 1

The Correct Option is C

Given \(A = \begin{pmatrix} 0 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{pmatrix}\), we need to find \(A^4\).

First, let's find \(A^2\):

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

Now, let's find \(A^4\):

\(A^4 = (A^2)^2 = I^2 = I \cdot I = I = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix}\)

Thus, \(A^4 = I\).

Therefore, the correct option is (C) I.

Answer 2

Given the matrix \( A \): \[ A = \begin{pmatrix} 0 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{pmatrix} \] \( A^4 \). \( A^2 \). \( A^2 = A \times A \): \[ A^2 = \begin{pmatrix} 0 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{pmatrix} \times \begin{pmatrix} 0 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{pmatrix} = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix} = I \] \( A^4 \). Since \( A^2 = I \), \( A^4 \): \[ A^4 = A^2 \times A^2 = I \times I = I \] Thus, \( A^4 = I \), the identity matrix.