Let A be a 3 3 matrix and assume that A can be transformed into a

Question:

Let A be a 3 × 3 matrix and assume that A can be transformed into a lower triangular matrix L using only column operations of type III; that is,
AE1E2E3, = L
where E1, E2, E3 are elementary matrices of type III. Let
U = (E1E2E3)-1
Show that U is upper triangular with l's on the diagonal and A = LU. (This illustrates a column version of Gaussian elimination.)