The idea of matrix computations is probably at least a century
old, I'm not sure it existed in the 19^{th} century already
much, and it has taken on a major scientific meaning since computers
have started to be used for scientific computations, lets say from
the 60 or 70's. Matrices are as the word suggests rows and columns
with numbers, basically to represent systems of equations, and can be
used in algebraic representations, that is one can do formula math
with them.

Suppose we have two linear equations with two unknowns, we could write the general form as:

a**x
**+ b**y
**= c

c**x
**+ d**y
**= e

These are linear because there are no factors like x squared or sine(x) in it, just a constant times the unknowns x and y, and additions of them. Given the multiplier constants and the right hand side constants, we can unsually solve the set of equations using highschool math, most straightforwardly by subtracting the top equation from the bottom one after having multiplied it such that the c component becomes zero. Then we have a single unknown in the bottom equation, which we then can solve easily. Backsubstituting that result in the upper equation gives another single variable equation, which can be easily inverted to find the second unknown.

I've made a diary page about such computations before, pleaste check the diary page list.

The idea of the such equations is that for instance we could use them to solve the problem of distributing light over some areas who emit and reflect light to eachother. Suppose we're able to derive a set of linear equations which contain the amount of light being transferred from one area to another, for instance we take variables x1 through x6 to represent the amount of light emitting from the ceiling, floor and walls of a room, and that we write down for each emiting surface on the right hand side how much light energy is emited from that surface , and on the left hand side how that light is distributed over the various areas.

If we assume the unknown vector x to be the desired effective light level of the various areas, we can by 'inverting' the matrix system of equations find the illumination of each surface of the scene as the components of the unknown vector (x,y,...). That leaves us with the problem of knowing how much light from each area lands on every other surface, which can be measured, and also computed from the physical properties of the areas and their relative orientation, though that is not trivial, because the light bounces back and forth between all surfaces after it came from a light source in principle infinetly, or until it has in the process been attenuated until negligable.