Definition of determinant of a matrix

Question

I have started studying linear algebra and I came across determinants. I know there are all sorts of formulae for computing it but I can't find what exactly it is. Why are they calculated the way there are? and is there a definition ?

Answer 1

I will briefly describe what a determinant is for a real matrix $A= [a_1$ $a_2$ ... $ a_n]$. Let $$ e_i(n) = \left\{ \begin{array}{ll} 1 & \quad n=i \\ 0 & \quad otherwise \end{array} \right. $$ Then $e_1,e_2,...e_n$ is a basis for $\mathbb{R}^n$. Note that $A(e_i)=a_i $ for $i=1..n$. So A maps unit n-cube to the n-dimensional parallelopiped defined by the vectors $a_1 , a_2 , … , a_n$ .

Loosely speaking the determinant is just the volume of this parallelopiped. In general sense, the determinant is the n-dimensional volume scaling factor of A.

Answer 2

In a vector space of dimension $n$ with (ordered) basis $\mathcal B=(e_1,\dots,e_n)$, one shows the space of alternate $n$-linear map has dimension $1$.

One takes as a generator the alternate form which takes the value $1$ on $\mathcal B$ (it is a function of the $n$ vectors), and call it the determinant map.

The determinant of $n$ vectors is then the value of the $\det$ map for this (ordered) list of vectors.