4 Determinants

When we computed the inverse of a \(2\times 2\) matrix \(A=\begin{pmatrix}a & b\\ c & d\end{pmatrix}\) we saw that it is invertible if \(ad-bc\neq 0\). This combination of numbers has a name, it is called the determinant of \(A\), \[\begin{equation} \det A = \det \begin{pmatrix}a & b\\ c & d\end{pmatrix}:=ad-bc. \tag{4.1}\end{equation}\] The determinant is a single number we can associate with a square matrix, and it is very useful, since many properties of the matrix are reflected in that number.

The determinant also has a geometric meaning: \(|\det A|\) is the volume of the parallelogram (or higher-dimensional equivalent) with edges that correspond to the column vectors of \(A\); in fact, \(\det A\) can be thought of as a “signed area (or volume)”. (For this reason determinants appear in multi-variable calculus.)

While determinants have many useful and simple properties, explicit formulas for determinants are more complicated. So in our treatment of determinants of \(n\times n\) matrices for \(n>2\) we will use an axiomatic approach, i.e., we will single out a few properties of the determinant and use these to define what a determinant should be, and then derive other properties from them. While this approach is conceptually clear, it has a slight disadvantage by being rather abstract at the beginning, before we eventually arrive at some explicit formulas. But along the way we will encounter some key mathematical ideas which are of wider use.