Triangular Factors, Row Exchanges, and Proper Matrix Notation

There are actually two reasons I am posting this question(Which has 3 parts)

1) Because it's really fun

2) Because I can't figure out parts 2 and 3.

Apply elimination to produce the factors L and U for

Part 1: A =

Part 2: A =

Part 3: A =

Now for the first one, what I did was(and I hope I get the notation on this right)

E = Elimination Matrix,

EA = This produced

Multiply by A, to get

U =

A=LU since this brought me back to A, I knew I was right...

Now when getting into the 3x3 matrices, I started getting into trouble.

Do I basically use a guess and check method? Or is there a way I can systematically solve this using elimination matrices and LU?

Are Row Exchanges necessary for part 2 and 3?

Can I avoid using Guess and Check for L and U?

Also I'm going to put what I did in Part 1 in Matrix Notation(or at least attempt to) and was wondering if someone could tell me if it is correct.

Which would produce