(a) Add and multiply the matrices

${\displaystyle \left(\begin{array}{cc}\hfill x\hfill & \hfill -y\hfill \\ \multicolumn{1}{c}{y}& \hfill x\hfill \end{array}\right)\hspace{0.5em}\mathrm{and}\hspace{0.5em}\left(\begin{array}{cc}\hfill u\hfill & \hfill -v\hfill \\ \multicolumn{1}{c}{v}& \hfill u\hfill \end{array}\right).}$

(b) What are the identities and inverses for addition and multiplication?

(c) Consider also the subset $R^{*}$ for which $y=0$. Investigate the arithmetic of $R^{*}$ (addition, subtraction, multiplication, division, identities, inverses, distributive law) and compare it with the arithmetic of real numbers.

(d) Compare the arithemetic of $C^{*}$ with that of complex numbers.

(e) The matrix $\left(\begin{array}{cc}\hfill -1\hfill & \hfill 0\hfill \\ \multicolumn{1}{c}{0}& \hfill -1\hfill \end{array}\right)$ is equivalent to the real number -1. What is the matrix equivalent to $i=\sqrt{-1}$ in the set of complex numbers?