Question

If A =



(i 0

0 i)

, where i =

√−1, find A^2

and A^4

(Matrices)

Answer 1

$$A=\begin{pmatrix} i & 0\\ 0 & i \end{pmatrix}$$

$$A^2=\begin{pmatrix} i & 0\\ 0 & i \end{pmatrix}\begin{pmatrix} i & 0\\ 0 & i \end{pmatrix}=$$

$$=\begin{pmatrix} i^2+0 & 0+0\\ 0+0 & 0+i^2 \end{pmatrix}=\begin{pmatrix} -1 & 0\\ 0 & -1 \end{pmatrix}$$

$$A^4=\begin{pmatrix} -1 & 0\\ 0 & -1 \end{pmatrix}\begin{pmatrix} -1 & 0\\ 0 & -1 \end{pmatrix}=$$

$$=\begin{pmatrix} (-1)^2+0 & 0+0\\ 0+0 & 0+(-1)^2 \end{pmatrix}=\begin{pmatrix} 1 & 0\\ 0 & 1 \end{pmatrix}$$