Chapter 6, Problem 5CTB

Let
$A=\left(\begin{array}{ccc}4& 2& 2\\ 2& 10& 10\\ 2& 10& 14\end{array}\right)$

1. Without computing the eigenvalues of A, show that A is positive definite.
2. Factor A into a product $LD{L}^T$ where L is unit lower triangular and D is diagonal.
3. Compute the Cholesky factorization of A.