- Calculus, Volume 2: Multi-Variable Calculus and Linear Algebra with Applications to Differential Equations and Probability
- Tom M. Apostol
- Second Edition
- 1991
- 978-1-119-49676-2
3.1 $\quad$ Introduction
$\quad$ Determinants of order two and three were introduced in Volume 1 as a useful notation for expressing certain formulas in a compact form. We recall that a determinant of order two was defined by the formula \begin{align*} (3.1) \qquad \begin{vmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{vmatrix} &= a_{11}a_{22} - a_{12}a_{21}. \end{align*} Despite the notational similarity to the $2 \times 2$ matrix $\begin{bmatrix}a_{11} & a_{12} \\ a_{21} & a_{22}\end{bmatrix},$ the determinant is a number assigned to the matrix according to (3.1). To emphasize the distinction, we may also write \begin{align*} \begin{vmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{vmatrix} &= \det\begin{bmatrix}a_{11} & a_{12} \\ a_{21} & a_{22}\end{bmatrix} \end{align*}
$\quad$ Determinants of order three were defined in Volume 1 in terms of second-order determinants by the formula
\begin{align*}
(3.2) \qquad
\det\begin{bmatrix}
a_{11} & a_{12} & a_{13} \\
a_{21} & a_{22} & a_{23} \\
a_{31} & a_{32} & a_{33}
\end{bmatrix}
&=
a_{11}\begin{vmatrix}
a_{22} & a_{23} \\ a_{32} & a_{33}
\end{vmatrix}
-
a_{12}\begin{vmatrix}
a_{21} & a_{23} \\ a_{31} & a_{33}
\end{vmatrix}
+
a_{13}\begin{vmatrix}
a_{21} & a_{22} \\ a_{31} & a_{32}
\end{vmatrix}
\end{align*}
This chapter treats the more general case, the determinant of a square matrix of order $n$ for any integer $n \geq 1.$ Our point of view is to treat the determinant as a function which assigns to each square matrix $A$ a number called the determinant of $A$, denoted by $\det A.$ It is possible to define this function as a generalized version of (3.1) and (3.2), but this formula is a sum of $n!$ products of the entries of $A,$ which may become unwieldy in practice. Instead, we can study determinants in terms of essential properties which we can deem as axioms. We will determine these properties in three stages:
(1) $\quad$ Motivating the choice of axioms.
(2) $\quad$ Deducing further properties of determinants from these axioms.
(3) $\quad$ Proving that there is one and only one function that satisfies these axioms.