Find the Cholesky factorizations of the following matrices:
(a)
$$
\begin{aligned}
& \left(\begin{array}{rr}
3 & -2 \\
-2 & 2
\end{array}\right), \\
& \text { (e) }\left(\begin{array}{llll}
2 & 1 & 0 & 0 \\
1 & 2 & 1 & 0 \\
0 & 1 & 2 & 1 \\
0 & 0 & 1 & 2
\end{array}\right) .
\end{aligned}
$$
(b) $\left(\begin{array}{rr}4 & -12 \\ -12 & 45\end{array}\right)$,
(c) $\left(\begin{array}{rrr}1 & 1 & 1 \\ 1 & 2 & -2 \\ 1 & -2 & 14\end{array}\right)$,
(d) $\left(\begin{array}{lll}2 & 1 & 1 \\ 1 & 2 & 1 \\ 1 & 1 & 2\end{array}\right)$,
(e) $\left(\begin{array}{llll}2 & 1 & 0 & 0 \\ 1 & 2 & 1 & 0 \\ 0 & 1 & 2 & 1 \\ 0 & 0 & 1 & 2\end{array}\right)$.