[[the-gaussian-elimination|Gaussian elimination]] solves one system — but if you have to solve with the SAME matrix again and again (new right-hand sides), why redo the work? LU factors the matrix ONCE into a Lower-triangular times an Upper-triangular piece; then each new solve is two cheap triangular sweeps. Factor once, solve forever. It’s how real solvers handle many loads on one structure. Slide to factor and read off L and U.
LU decomposition factors a matrix A=LU into a unit-LOWER-triangular L (the elimination multipliers) and an UPPER-triangular U (the reduced matrix) — the bookkeeping of [[the-gaussian-elimination|Gaussian elimination]] made reusable. Factoring costs O(n³) ONCE; then each solve Ax=b is a forward sweep Ly=b and a back sweep Ux=y, only O(n²). So many right-hand sides on the same A (and the determinant ∏uᵢᵢ, and the inverse) come cheap. Partial pivoting gives PA=LU for stability. It is the standard dense direct solver (LAPACK). A fail-loud self-check throws unless L·U reproduces A. ◆ real numerical methods, node-verified.
Needs pivoting (PA=LU) for stability, and dense O(n³) factoring is costly for very large sparse systems (sparse/iterative methods win). The L·U=A factorization and reuse economics are exact.