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Mathematics

Cramer's Rule: Solving Linear Systems with Determinants

Quick fact

Cramer's rule can solve a 2x2 system almost instantly: for a system like ax + by = e, cx + dy = f, the solution is x = (ed - bf) / (ad - bc) and y = (af - ec) / (ad - bc). This rule extends to any square system, but it becomes computationally heavy for large systems—for a 100x100 system, it would require 101 determinant calculations, each of which is far more expensive than Gaussian elimination.

Why this is interesting

You've probably solved systems of equations by elimination, but what if you could just read the solution from a few numbers? Cramer's rule offers a surprisingly direct formula—but why does it work?