Determinants

For the goals of this class, the topic is a bit controversial. On one hand, determinants have great theoretical importance in linear algebra, they appear (and are very useful) in many other branches of mathematics, and have fascinating properties. On the other hand, like matrix inverses, they are almost completely irrelevant when it comes to large scale applications and practical computations (too expensive to compute). So we will discuss them very briefly, mostly stating their properties which we will need for later developments.

The determinant of a square matrix $A$ is a scalar, denoted $detA$ or $|A|$. There are many equivalent ways to define determinants. Let us start with "axiomatic" definition, which states several properties of the determinants, which uniquely define them, but do not tell us how to compute $detA$.

Definition

A determinant is a function $det:{\mathbb{M}}_{n\times n}\to \mathbb{R}$ such that:

1. $det{I}_n=1$
2. Is linear in each row:$$\det\begin{bmatrix}- & a^1 & - \& \vdots & \ -&a^{i-1}&-\-&\alpha b+\beta c&-\-&a^{i+1}&-\ &\vdots& \-&a^n&-\end{bmatrix}=\alpha\det\begin{bmatrix}- & a^1 & - \& \vdots & \ -&a^{i-1}&-\-&b&-\-&a^{i+1}&-\ &\vdots& \-&a^n&-\end{bmatrix}+\beta\det\begin{bmatrix}- & a^1 & - \& \vdots & \ -&a^{i-1}&-\-&c&-\-&a^{i+1}&-\ &\vdots& \-&a^n&-\end{bmatrix}$$
3. Is an "alternating" function: $$\det\begin{bmatrix}-&a^1&-\&\vdots&\end{bmatrix}$$