On the contrary, that is precisely the "L" in the "LU" form.
Here's how I would get L: 'zero' out the area above the diagonal using row operations and keeping track of the row operations.
We can change to
by "subtract 3 times the first row from the second" and "subtract 1 times the first row from the third"
We can change that matrix to
in "upper triangular" form by "add the second row to the third row".
Now, do the opposite operations, in the opposite order, to the identity matrix.
That is, first "subtract the second row from the third row" to get
and then "add 1 times the first row to the third" and "add 3 times the first row to the second" to get
Now you can check that
or LU= A.