LU and Cholesky Decomposition

The Problem

1. LU decomposition. For

$A=\left(\begin{array}{ccc}2& 1& 1\\ 4& 3& 3\\ 8& 7& 9\end{array}\right),$

find a lower-triangular $L$ (with 1's on the diagonal) and an upper-triangular $U$ such that $A=LU$.

1. Cholesky decomposition. For the symmetric positive definite matrix

$B=\left(\begin{array}{ccc}4& 2& 2\\ 2& 5& 4\\ 2& 4& 6\end{array}\right),$

find a lower-triangular $L$ with positive diagonal entries such that $B=L{L}^T$.

1. State concisely when each decomposition exists and when one is preferred over the other.