To do this problem you only need to know how to add and multiply two by two matrices. The problem gives models for complex numbers and 4-dimensional numbers called quaternions. Although you don't need the information to do this problem you may like to read the NRICH article What are Complex Numbers? and the Plus article:Curious Quaternions .

(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?

(2) Complex numbers are two-dimensional numbers $x+\mathrm{iy}$ where $x$ and $y$ are real numbers and $i=\sqrt{-1}$. Quaternions are four-dimensional numbers of the form $a+ix+jy+kz$ where $a,x,y,z$ are real numbers and $i,j,k$ are all different square roots of $-1$.

Can such a number system exist? To answer this, let's assume that in part (1) you have established that if real numbers and 2 by 2 matrices exist then so does the complex number $i=\sqrt{-1}$. The model for the system of quaternions is the set of linear combinations of 2 by 2 matrices:

${\displaystyle \mathrm{aI}+ix+jy+kz\hspace{0.5em}=a\left(\begin{array}{cc}\hfill 1\hfill & \hfill 0\hfill \\ \multicolumn{1}{c}{0}& \hfill 1\hfill \end{array}\right)+x\left(\begin{array}{cc}\hfill i\hfill & \hfill 0\hfill \\ \multicolumn{1}{c}{0}& \hfill -i\hfill \end{array}\right)+y\left(\begin{array}{cc}\hfill 0\hfill & \hfill 1\hfill \\ \multicolumn{1}{c}{-1}& \hfill 0\hfill \end{array}\right)+z\left(\begin{array}{cc}\hfill 0\hfill & \hfill i\hfill \\ \multicolumn{1}{c}{i}& \hfill 0\hfill \end{array}\right).}$

where $a,x,y,z$ are real numbers.

(a) Work out the matrix products: $i^2,\hspace{0.5em}{j}^2$ and $k^2$ showing that these matrices give models of three different square roots of -1.

(b) Work out the matrix products: $ij$, $ji$, $jk$, $kj$, $ki$ and $ik$ showing that the three matrices $i,\hspace{0.5em}j,\hspace{0.5em}k$ model unit vectors along the three axes in 3-dimensional space $R^3$ with matrix products isomorphic to vector products in $R^3$.

You have now shown that this set of linear combinations of matrices models the quaternions.

(3) Investigate the sequence: $i,\hspace{0.5em}$ $ij,\hspace{0.5em}$ $ij$ $k,\hspace{0.5em}$ $ij$ $ki,\hspace{0.5em}$ $ij$ $ki$ $j,\hspace{0.5em}...$

... and the story continues in the Plus article: Ubiquitous Octonions.