TITLE: History of the matrix representation of complex numbers
QUESTION [14 upvotes]: It is well-known to many that $\mathbb{C}$ can be represented by matrices of the form $\left[ \begin{array}{cc} a & b \\ -b & a \end{array} \right]$. For example, see this question or this question. It is also discussed in the wikipedia article history of complex numbers article. Apparently, there is even an introductory complex variable textbook by Copson from 1935 which uses such matrices to define complex numbers. This is mentioned in Numbers by Ebbinghaus et. al. on page 69.
My question is simply this:

What is the history of this construction? Who first explained that complex numbers could be viewed as $2 \times 2$ matrices of the special form $\left[ \begin{array}{cc} a & b \\ -b & a \end{array} \right]$ ?

I realize this is just the regular representation of $\mathbb{C}$, and I realize such matrices are the matrices of a dilation composed with a rotation and possibly a reflection, but, the question still remains, who did found these first? References are appreciated.

REPLY [3 votes]: Today was the first day of class in my complex analysis course. I sometimes attempt new derivations in real time to keep it fresh. Today, we got to the point of asking what was the reciprocal of $z=x+iy$. We said, let $w=a+ib$ and seek solutions of $wz=1$. This gives:
$$ wz = (a+ib)(x+iy) = ax-by+i(bx+ay) = 1+i(0).$$
Equating real and imaginary parts reveals:
$$ ax-by = 1 \qquad \& \qquad bx+ay = 0 $$
which is a system of linear equations which has matrix form:
$$ \left[ \begin{array}{cc} x & -y \\ y & x \end{array}\right]\left[ \begin{array}{c} a \\ b \end{array}\right] =\left[ \begin{array}{c} 1 \\ 0 \end{array}\right]  $$
We solve for $[a,b]^T$ by multiplying by the inverse of the $2 \times 2$ matrix for which we have the handy-dandy formula $\displaystyle \left[ \begin{array}{cc} x & -y \\ y & x \end{array}\right]^{-1} = \frac{1}{x^2+y^2}\left[ \begin{array}{cc} x & y \\ -y & x \end{array}\right]$. Thus,
$$ \left[ \begin{array}{c} a \\ b \end{array}\right] = \frac{1}{x^2+y^2}\left[ \begin{array}{cc} x & y \\ -y & x \end{array}\right]\left[ \begin{array}{c} 1 \\ 0 \end{array}\right] = \frac{1}{x^2+y^2}\left[ \begin{array}{c} x \\ -y \end{array}\right].$$
Therefore, $a = \frac{x}{x^2+y^2}$ and $b = \frac{-y}{x^2+y^2}$ so
$$\frac{1}{z}= \frac{x-iy}{x^2+y^2}.$$
I just found this a nice illustration of JHance's comment. In this routine calculation we stumble upon the $2 \times 2$ representations of both $z=x+iy$ and $1/z$. So, perhaps the real question to ask is not when the matrix representation was first given. Rather, the real question is simply when was the algebra of small matrices first known. I gather from yoknapatawpha's post of the 1858 paper of Cayley it may be a few years before that work. Apparently, the term matrix is Latin for "womb" and is due to Sylvester in 1850 as you may read at history of matrices. This makes me think there may exist some improvement on the Cayley answer.