LU factorization

$A = LU$ — elimination's multipliers ( $L$ ) times the resulting triangular matrix ( $U$ ).

When Gaussian elimination reduces a square $A$ to upper triangular $U$ using only row-combination operations (no swaps), collecting every multiplier $m_{ik}$ used into a unit lower-triangular matrix $L$ gives $A = LU$ . This falls directly out of each elimination step being left multiplication by an elementary matrix.