Gaussian Elimination and Row Echelon Form

Every risk engine that solves a linear system — a replicating portfolio, a hedge ratio, a set of no-arbitrage prices implied by traded instruments — ultimately runs some variant of the algorithm in this note. Gaussian elimination systematically removes unknowns from a system, one at a time, until it is triangular and trivial to solve. This note builds the algorithm from the row picture up, and — the conceptual payoff — shows that each elimination step is itself a matrix multiplication, setting up the $A=LU$ factorisation of the derivation note that follows.

Learning objectives. By the end of this note you should be able to: