Answer
See the step-by-step work.
Work Step by Step
a)
We multiply the matrices as follows:
$\begin{array}{l}
A = \left[ {\begin{array}{*{20}{c}}
i&0\\
0&i
\end{array}} \right]\\
{A^2} = A.A\\
= \left[ {\begin{array}{*{20}{c}}
i&0\\
0&i
\end{array}} \right].\left[ {\begin{array}{*{20}{c}}
i&0\\
0&i
\end{array}} \right]\\
= \left[ {\begin{array}{*{20}{c}}
{{i^2}}&0\\
0&{{i^2}}
\end{array}} \right] = \left[ {\begin{array}{*{20}{c}}
{ - 1}&0\\
0&{ - 1}
\end{array}} \right]\\
{A^3} = {A^2}.A\\
= \left[ {\begin{array}{*{20}{c}}
{{i^2}}&0\\
0&{{i^2}}
\end{array}} \right].\left[ {\begin{array}{*{20}{c}}
i&0\\
0&i
\end{array}} \right]\\
= \left[ {\begin{array}{*{20}{c}}
{{i^3}}&0\\
0&{{i^3}}
\end{array}} \right]\\
= \left[ {\begin{array}{*{20}{c}}
{ - i}&0\\
0&{ - i}
\end{array}} \right]
\end{array}$
Observe that the diagonal entries of $A^n$ correspond to $i^n$
b)
\begin{array}{l}
{B^2} = B.B\\
= \left[ {\begin{array}{*{20}{c}}
0&{ - i}\\
i&0
\end{array}} \right].\left[ {\begin{array}{*{20}{c}}
0&{ - i}\\
i&0
\end{array}} \right]\\
= \left[ {\begin{array}{*{20}{c}}
{ - {i^2}}&0\\
0&{ - {i^2}}
\end{array}} \right] = \left[ {\begin{array}{*{20}{c}}
1&0\\
0&1
\end{array}} \right]
\end{array}
