Definition of determinant of a matrix 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 ?
 A: 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.
A: 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.
