What's the difference between determinants and matrices? This question occurred in my mind after reading this theorem:A factor common to all elements of a row (or column) can be taken out as a factor of the determinant. In matrix,a factor common to ALL elements only can be taken out because then if you multiply back,you multiply all elements of the matrix with the same factor.This is certainly not the case for determinant.Also a matrix is just a system of equations.Then what is determinant?
 A: The determinant is actually a function $\det: \mathbb{R}^{n\times n}\rightarrow \mathbb{R}: A\mapsto \det(A)$. So to each square matrix we can assign a real number. So clearly matrices and determinants are completely different.
One of the most important features of a determinant is the following theorem:
A square matrix $A$ is invertible if and only if $\det(A)\neq 0.$
So in particular, the system of equations $AX=b$ has a unique solution if and only if $A$ is invertible (indeed, then $X=A^{-1}b)$. In order to check whether such a system has a unique solution it suffices to check that $\det(A)\neq 0$ which is easier than solving the system explicitly.
Calculating a determinant is another story, there are many rules for calculating a determinant, for example $\det(AB)=\det(A)\det(B)$ and many others.
A: The determinant is a multilinear function of the column of the matrix. This justify the theorem that you refer.
The determinant represents the oriented volume of the parallelepiped formed by the column vectors of the matrix.
And note that a matrix is not only ''a system of linear equations'' ( in the sense that we can write the system in matrix notation), but it represents a linear transformation between vector spaces.
