Question

3. If A = i 0

0 i


where i =√−1, find A^2

and A^4.

Answer 1

We know that A is

$A= \begin{pmatrix} i & 0 \\ 0 & i \end{pmatrix}$ . Therefore, $A^2 = \begin{pmatrix} i & 0 \\ 0 & i \end{pmatrix} \cdot \begin{pmatrix} i & 0 \\ 0 & i \end{pmatrix} = \begin{pmatrix} i\cdot i+0\cdot0 & i\cdot0 +0\cdot i\\ 0\cdot i + i\cdot0 & 0\cdot0+i\cdot i \end{pmatrix} = \begin{pmatrix} -1 & 0 \\ 0 & -1 \end{pmatrix}.$

Next, $A^4 = A^2\cdot A^2 = \begin{pmatrix} -1 & 0 \\ 0 & -1 \end{pmatrix} \cdot \begin{pmatrix} -1 & 0 \\ 0 & -1 \end{pmatrix} = \begin{pmatrix} -1\cdot-1 + 0\cdot0 & -1\cdot0+0\cdot-1 \\ 0\cdot-1+-1\cdot0 & 0\cdot0+-1\cdot-1 \end{pmatrix} = \begin{pmatrix} 1 & 0 \\ 0 &1 \end{pmatrix}$

We may also note that $A = i\cdot \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix},$ therefore $A^2 = (-1)\cdot\begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix},$ $A^4 = 1\cdot \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}.$