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Why do we use Gaussian Elimination when inverting a matrix? (this is only one of the methods...which is the only one that doesn't make sense to me).

I know how to do it, but I'm not sure why it works. When solving a system of linear equations, I understand why Gaussian Elimination works: to me, it's just the adding and subtracting of equations until a desirable form is reached. But Gaussian Elimination doesn't seem to be as obvious a tool in matrix inversion.

Thanks!