After factoring A = LU, solving Ax = b takes:
LU Factorization
Factoring A = LU (unit lower triangular L, echelon U), solving Ly = b then Ux = y, and the algorithm that builds L from the row-replacement multipliers.
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Matrix Factorization
Notes
A = LU with L unit lower triangular and U an echelon form of A. Solve Ax = b by Ly = b (forward substitution), then Ux = y (back substitution).
Exam angle
Example:
An LU Factorization Algorithm
Notes
Row reduce A to U using only row REPLACEMENTS (adding multiples of rows above); L stores the multipliers with opposite sign below its unit diagonal.
Memory trick If R2 → R2 − 3R1, put +3 in L at position (2, 1).