Why it matters

Gaussian elim solves systems + finds inverse + rank + determinant. Multi-purpose.

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The architecture

Forward: reduce to upper triangular.

Back: substitute for solution.

Pivoting: stability.

Gaussian eliminationForward elimupper triangularBack subsolve variablesPivotfor stabilityPartial pivoting: swap rows to largest pivot for numerical stability
Steps.
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How it works end to end

Partial pivoting: swap rows to place largest pivot on diagonal.

Full pivoting: also swap columns.

Back substitution O(n^2).