Will the matrices always be 2 x 2? If so, there's a known, explicit inversion formula for that. (Google for it, or look on places like
MathWorld or
Planet Math.) You can figure it out yourself on paper very easily, too.
I'm going to use the matlab-like notation [a,b;c,d] to represent the matrix
since it's a bit hard to get the formatting just right.
By the definition of an inverse matrix, you need
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[a,b;c,d] [e,f;g,h] = [1,0;0,1]
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Do the matrix multiplication:
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[ae+bg,af+bh;ce+dg,cf+dh] = [1,0;0,1]
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Two matrices are equal iff all the corresponding entries are equal. So,
ae + bg = 1
af + bh = 0
ce + dg = 0
cf + dh = 1
Four equations in four unknowns. You just need to use some simple algebra from here.
For larger matrices, I'd recommend googling for "Gaussian elimination with partial pivoting." That algorithm is usually used to solve the system Ax = b by adjoining the matrices like [A | b], but if you use an identity matrix I in place of b and adjoin [A | I], then you'll get the inverse of A when you're done. GEPP is the way to keep track of all the operations that were suggested to you by the previous poster. If your matrix system turns out to be ill-conditioned (poorly behaved), then "Gaussian elimination with full pivoting" (GEFP) may be in order. In practice, I've never encountered small matrix systems that needed full pivoting.
For even larger matrices, note that it's not computationally practical to find the full inverse matrix (nor even desirable). If you want to solve Ax = b for large matrices A, then you'll need to use iterative techniques (which give approximations to the solution that converge as you continue to iterate) like conjugate gradients (if A is symmetric and positive definite), biconjugate gradient, successive overrelaxation (SOR), Jacobi iteration, Gaussi-Seidel, etc. I routinely have to solve systems of approximately 250,000 variables for my cancer research on desktop systems, and it would be a non-starter to invert the matrix, particularly given that we need to solve several such systems per simulated time step (aproximately one simulated hour), for a total simulation of 30-180 virtual days.
I hope you enjoy your project--matrix inversion sounds simple, but is actually a rich research topic in and of itself, and it forms a major part of the foundation of scientific computing. (To solve partial differential equations, you just end up discretizing the equation and writing it as a system of linear equations. Voila: you've translated a differential equation problem into a massive linear algebra problem.) -- Paul
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