There is a bit of a pedagogical problem when introducing matrices and determinants. They are very useful tools and there are very good reasons to define them the way we do. But it is typically not immediately possible to tell the students about these reasons.
For multiplication of matrices, one preliminary reason could be that it gives us an easier notation to denote systems of linear equations. But the real reason of course is that multiplication of matrices come from a very geometrical object, namely the linear maps. Once you understand linear maps, you'll be convinced of the necessity of the matrix multiplication.
The determinant is a very complicated expression. But it essentially has one primary use, namely that it decides how many solutions a system of equations have. Or equivalently, it decides whether a matrix is invertible or not. There are many geometrical ways of seeing the determinant, a favorite one is that the determinant gives the volume of a parallelepiped.
A good introductory book (with proofs) is "Introduction to Linear Algebra" by Lang (be sure not to get his more advanced "Linear algebra").
https://www.amazon.com/dp/0387962050/?tag=pfamazon01-20
A very very good book (but also more advanced) is Treil's "Linear algebra done wrong":
http://www.math.brown.edu/~treil/papers/LADW/LADW.html It is completely free too.
