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Question: If A = \(\begin{bmatrix} i & 0 \\ 0 & i \end{bmatrix}\), n Î N . Then A<sup>4n</sup> equals...

If A = [i00i]\begin{bmatrix} i & 0 \\ 0 & i \end{bmatrix}, n Î N . Then A4n equals

A

[1001]\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}

B

[i00i]\begin{bmatrix} i & 0 \\ 0 & i \end{bmatrix}

C

[0ii0]\begin{bmatrix} 0 & i \\ i & 0 \end{bmatrix}

D

[0000]\begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix}

Answer

[1001]\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}

Explanation

Solution

A2 = A.A = [i00i]\begin{bmatrix} i & 0 \\ 0 & i \end{bmatrix} = $\begin{bmatrix}

  • 1 & 0 \ 0 & - 1 \end{bmatrix}$

A4 = A2A2 = [−100−1]\left[ \begin{array} { c c } - 1 & 0 \\ 0 & - 1 \end{array} \right] $\begin{bmatrix}

  • 1 & 0 \ 0 & - 1 \end{bmatrix}== \begin{bmatrix} 1 & 0 \ 0 & 1 \end{bmatrix}$ = I

A4n = In = I