Where does matrix multipication come from?

Hi,

I am trying to get my head around the origin or matrix multiplication.

Is a good description just: "Matrix Multiplication is just solving a system of linear equations. For example a nxm matrix 'A' (representing the coficients of n equations with m coefficients each) 'multiplies' by a mx1 vector 'V' (representing the variables), and the multiplication process is just solving for the variable; except in matrix multiplcation the the enries in V are actually constants and the anser vector holds the 'variables'. In the case when the vector V has more than one column this is just a a range of cases with different values for the enries in V. So, when V is a matrix, the answer to A x V, is a matrix with the number of columns equal to the number of cases set out as columns in the matrixs V." I just wrote this from the top of my head.

I have been thinking about matrix multiplication like a mxnxk box where matrix A is mxn and matrix be is nxk. The front of the box is matrix A and the top is matrix B. Various points inside the box coincide with entries in A and B, and the values at these points are the product of the coresponding enrtries in a A and B. 'Surfaces' A and B of the box are joined along the side 'n' long. Summing along the n direction give values in a mxk matrix which is tyhe side of the box. This side is the answer of matirx multiplication.

What is the role of determonants in matrix multiplication? I read in wikipedia that the determinant of a nxn matrix can be used to represent the volume inside a rhombus in n-space whose sides are made up of the entries in the matrix. Haven't thought about this enough yet.

My question is can anyone give me some pointers?

Thanks,

Ed