This question is from Artin Algebra, Ch 2, Exercise M11(d):
Most invertible matrices can be written as a product $A=LU$ of a lower triangular matrix $L$ and an upper triangular matrix $U$, where in addition all diagonal entries of $U$ are $1$.
Describe the double cosets $LgU$.
In fact complete exercise is this:
where Exer M.9 is this:
I could do this only: $LIU=A$.
Also, does there exist unique matrices $P,L',D,U$ for any invertible matrix $A$, such that $PA=L'DU$, where $P$ is permutation matrix, $L'$ is lower triangular matrix with all diagonal entries $1$, $D$ is a diagonal matrix and $U$ (as above,) upper triangular matrix with all diagonal entries $1$. ?