Chapter 8.2, Problem 92E

a.

To determine

To find: $A^2$ , $A^3$ and $A^4$

Answer to Problem 92E

  $\begin{array}{l}{A}^2=\left[\begin{array}{cc}-1& 0\\ 0& -1\end{array}\right]\\ A^3=\left[\begin{array}{cc}-i& 0\\ 0& -i\end{array}\right]\\ A^4=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]\end{array}$

  $A^2$ is similar to $i^2$ .

  $A^3$ is similar to $i^3$ .

  $A^4$ is similar to $i^4$ .

Explanation of Solution

Given:

  $A=\left[\begin{array}{cc}i& 0\\ 0& i\end{array}\right]$

It is known that,

  $i=\sqrt{-1}$

So,

  $i^2=-1$

  $i^3=-i$

  $i^4=1$

Now consider,

  $\begin{array}{c}{A}^2=AA\\ =\left[\begin{array}{cc}i& 0\\ 0& i\end{array}\right]\left[\begin{array}{cc}i& 0\\ 0& i\end{array}\right]\\ =\left[\begin{array}{cc}{i}^2+0& 0+0\\ 0+0& 0+i^2\end{array}\right]\end{array}$

  $\begin{array}{c}=\left[\begin{array}{cc}{i}^2& 0\\ 0& i^2\end{array}\right]\\ =\left[\begin{array}{cc}-1& 0\\ 0& -1\end{array}\right]\end{array}$

  $\begin{array}{c}{A}^3=A^{2}A\\ =\left[\begin{array}{cc}-1& 0\\ 0& -1\end{array}\right]\left[\begin{array}{cc}i& 0\\ 0& i\end{array}\right]\\ =\left[\begin{array}{cc}-i+0& 0+0\\ 0+0& 0-i\end{array}\right]\\ =\left[\begin{array}{cc}-i& 0\\ 0& -i\end{array}\right]\end{array}$

  $\begin{array}{c}{A}^4=A^{3}A\\ =\left[\begin{array}{cc}-i& 0\\ 0& -i\end{array}\right]\left[\begin{array}{cc}i& 0\\ 0& i\end{array}\right]\\ =\left[\begin{array}{cc}-i^2+0& 0+0\\ 0+0& 0-i^2\end{array}\right]\end{array}$

  $\begin{array}{c}=\left[\begin{array}{cc}-i^2& 0\\ 0& -i^2\end{array}\right]\\ =\left[\begin{array}{cc}-\left(-1\right)& 0\\ 0& -\left(-1\right)\end{array}\right]\\ =\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]\end{array}$

So, it can be said that, $A^2$ is similar to $i^2$ , $A^3$ is similar to $i^3$ and $A^4$ is similar to $i^4$ .

b.

To determine

To find: $B^2$

Answer to Problem 92E

  $B^2=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$

Explanation of Solution

Given:

  $B=\left[\begin{array}{cc}0& -i\\ i& 0\end{array}\right]$

  $\begin{array}{c}{B}^2=BB\\ =\left[\begin{array}{cc}0& -i\\ i& 0\end{array}\right]\left[\begin{array}{cc}0& -i\\ i& 0\end{array}\right]\\ =\left[\begin{array}{cc}0-i^2& 0+0\\ 0+0& -i^2+0\end{array}\right]\end{array}$

  $\begin{array}{c}=\left[\begin{array}{cc}-i^2& 0\\ 0& -i^2\end{array}\right]\\ =\left[\begin{array}{cc}-\left(-1\right)& 0\\ 0& -\left(-1\right)\end{array}\right]\\ =\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]\\ =A^4\end{array}$