Question

If A =



(i 0

0 i)

, where i =

√−1, find A^2

and A^4

(Matrices)

Answer 1

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

$A^4=\begin{pmatrix} -1 & 0 \\ 0 & -1 \end{pmatrix}\begin{pmatrix} -1 & 0 \\ 0 & -1 \end{pmatrix}=\begin{pmatrix} -1(-1)+0*0 & -1*0+0(-1) \\ 00(-1)-1*0 & -1(-1)+0*0 \end{pmatrix}=\begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}$