If A = [(0,i)(i,0)], n ∈ N, the A^(4n) equals : ​

Question

If A =\(\begin{bmatrix} 0 & i \\[0.3em] i & 0 \\[0.3em] \end{bmatrix}\), n ∈ N, the A⁴ⁿ equals :

A. \(\begin{bmatrix} 0 & i \\[0.3em] i & 0 \\[0.3em] \end{bmatrix}\) 

B. \(\begin{bmatrix} 0 &0 \\[0.3em] 0 & 0 \\[0.3em] \end{bmatrix}\) 

C. \(\begin{bmatrix} 1 & 0 \\[0.3em] 0 & 1 \\[0.3em] \end{bmatrix}\) 

D. \(\begin{bmatrix} 0 & i \\[0.3em] i & 0 \\[0.3em] \end{bmatrix}\)

Answer 1

(C). \(\begin{bmatrix} 1 & 0 \\[0.3em] 0 & 1 \\[0.3em] \end{bmatrix}\) 

A⁴ⁿ = A⁴

= \(\begin{bmatrix} 0 & i \\[0.3em] i & 0 \\[0.3em] \end{bmatrix}\) x \(\begin{bmatrix} 0 & i \\[0.3em] i & 0 \\[0.3em] \end{bmatrix}\) x \(\begin{bmatrix} 0 & i \\[0.3em] i & 0 \\[0.3em] \end{bmatrix}\) x \(\begin{bmatrix} 0 & i \\[0.3em] i & 0 \\[0.3em] \end{bmatrix}\) 

{n = 1, so the exponent comes out to be 4 and if n = 2, which will turn the exponent to 8, and the same cycle will repeat.}

Option (C) is the answer.