"Does solution to LU factorization needed to be unique for each question? Can a question have many solutions? (Many combinations of L and U) Ex. 1 Question 18 c)"

In general there are many LU factorisations possible and the one you sent me is perfectly valid.

You can verify your factorisation by multiplying the matrices and you do in fact get the original matrix. Sure enough, your matrices are upper/lower triangular so it is valid.

There is an additional condition that we may set which will guarantee a unique solution. If we say that the diagonal entries of one of the matrices must be equal to 1 then there is a theorem that the factorisation will be unique.

However we will not make use of this in our course since it suffices to find any LU factorisation in order to solve a system of linear equations.