$4\mathrm{.}5$ P$-$L$-$U Factorization Method, and re$-$solving the problem for a new $b$ vector

a$)$ Use the results of your G$-$E process in $4\mathrm{.}4$ to manually construct the $P-L-U$ matrices for the $A$

matrix in the problem above. Show your work! You're being graded on demonstrating your

$+4$ understanding between the G$-$E process and the P$-$L$-$U matrices, but you can catch mistakes by

comparing your work to the output from the MATLAB $"$lu$"$ command, and$/$or checking $PA=LU.$

$+7$ b$)$ Use your $P-L-U$ factorization of A from $($a$)$ to

"efficiently" solve $($by$-$hand!$)$ the new problem at

right:

The point of $($b$)$ is to recognize the new problem has

the same A matrix, so you must NOT go through the

$\left[\begin{array}{ccc}2& 2\mathrm{.}2& -3\mathrm{.}6\\ 10& 1& -3\\ -1& -8\mathrm{.}1& 4\mathrm{.}3\end{array}\right]([c],[u],[d])=([4],[-10],[-39])$

entire G$-$E

process again. Just use the P$-$L$-$U s from part $($a$)$ to more quickly solve $Ax=b$ for the new $b$ vector.