LU Decomposition Without Pivoting

[Source: Trefethen-Bau Numerical Linear Algebra]

Now, to exhibit the full factorization A = LU, we need to compute the product
L = L1-1 L2-1 L3-1 . Perhaps surprisingly, this turns out to be a triviality. The inverse of L1 is just L1 itself, but with each entry below the diagonal negated

The inverses of L2 and L3 are obtained similarly: just negate their sub diagonal entries. Finally, the product   L1-1 L2-1 L3-1   is just the unit lower-triangular matrix with the nonzero sub diagonal entries of L1-1, L2-1, and L3-1 inserted in the appropriate places. Thus