[翻譯]Arthur Cayley，矩陣理論紀要(A Memoir on the Theory of Matrices)，part: 2

The notion of such a matrix arises naturally from an abbreviated notation for a set of linear equations, viz. the equations
$$
\begin{eqnarray*}
X &=& ax+by+cz,\\
Y &=& a'x+b'y+c'z\\
Z &=& a''x+b''y+c''z
\end{eqnarray*}
$$
may be more simply represented by
$$
\left[
\begin{array}{c}
X \\
Y \\
Z
\end{array}
\right]
=
\left[
\begin{array}{ccc}
a & b & c \\
a' & b' & c' \\
a'' & b'' & c''
\end{array}
\right]
\left[
\begin{array}{c}
x \\
y \\
z
\end{array}
\right]
$$
and the consideration of such a system of equations leads to most of the fundamental notions in the theory of matrices.

矩陣的概念係由簡記一組線性方程式自然派生而出，所謂線性方程式，例如
$$
\begin{eqnarray*}
X &=& ax+by+cz,\\
Y &=& a'x+b'y+c'z\\
Z &=& a''x+b''y+c''z
\end{eqnarray*}
$$

可簡記為

$$
\left[
\begin{array}{c}
X \\
Y \\
Z
\end{array}
\right]
=
\left[
\begin{array}{ccc}
a & b & c \\
a' & b' & c' \\
a'' & b'' & c''
\end{array}
\right]
\left[
\begin{array}{c}
x \\
y \\
z
\end{array}
\right]
$$

而關於線性方程組的研究，引領出矩陣理論中最基本的幾個概念。