LU decomposition

Let $A$ be a 3x3 square invertible matrix. Elimination on $A$ by a sequence of elimination matrices $E_{21},E_{31}$ and $E_{32}$ results in an upper triangular matrix $U$:

We can combine the elimination matrices together:

and invert to arrive at the LU decomposition:

The elements of the matrix $L$ have a very direct interpretation, arguably more interpretable than those of $E$. If $l_{21},l_{31}$ and $l_{32}$ are the scalars by which the first and second rows are subtracted from the rows below, then we can express $L$ as: