What is the geometrical interpretation of determinant of a matrix in general? My question is simple (and maybe I am wrong  asking this question even) what is the geometrical interpretation of determinant of a matrix in general ? 
I could not think anything.
 A: In Lax's Linear Algebra and it's Applications, he furnishes an interpretation of determinants as closely related to the volume of a "simplex".
Kind of like how an n-cube is a generalization of a square and a cube, and an n-ball is a generalization of circles and spheres, a simplex is a generalization of triangles and pyramids. 
If you have a simplex of dimension $n$ and one of its sides lies in the space of the other sides, it's a "degenerate" simplex, and it has zero volume, just like that of a triangle and a pyramid. So, if you view the columns of your matrix as vectors, and one of your vectors is linearly dependent, the simplex constructable through the vectors is degenerate, so the matrix has determinant $0$.
If you calculate the volume of a triangle from a base lying on an "axis", but then decide to use a different side as the base, then you have to rotate it, which places it beneath the previous axis and turns the volume negative, just as how switching columns in a matrix causes the determinant to be negative.
Once you manage to wrap your head around of the columns of a matrix being the sides of a triangle/n-dimensional simplex, it generally makes the determinant make much more sense.
A: $n\times n$ matrix consist of $n$ vectors in $\Bbb{R}^n$. These $n$ vectors form a parallelogram, the determinant of the matrix is the volume of that parallelogram. For example, 
$$\begin{pmatrix} 1&0&0\\ 0&5&0\\ 0&0&9\end{pmatrix}$$
gives us three vectors in $\Bbb{R}^3$, one is on the $x$ axes of length $1$, the other is on the $y$ axes of length $5$ and the last is on the $z$ axes of length $9$. These three vectors defined a box of dimensions $1\times5\times9$, whose volume is equal to the determinant $45$.
