Let $P=\left[\begin{array}{ccc}2& -2& 6\\ 1& 3& 1\\ 4& 0& 2\end{array}\right]$

$($a$)(8$ points$)$ Find an $LU$ decomposition of $P,$ i$.$e$.$ find lower and upper triangular

matrices $L$ and $U$ such that $LU=P.$ Find this solution by hand in this part!

$($b$)(5$ points$)$ Recompute the $LU$ decomposition of $P$ using appropriate commands

in Matlab. Include your code and the results in your answer. Are both the lower $L$

and upper $U$ triangular factorisations returned by Matlab triangular as expected?

Comment on what you find!

$($c$)(5$ points$)$ Discuss the "cost" of computation of an $LU$ decomposition and its

subsequent use in solving a system of linear equations $Ax=b($ A and $b$ known,

solve for $x).$ Computing the $LU$ decomposition is typically quite a bit of work,

under what conditions might we expect it to be faster $/$ less computationally

costly than an alternative solution such as $A^{-1}Ax=Ix=A^{-1}b?$

$($d$)(7$ points$)$ Write a Matlab routine to compute and plot the difference in time it

takes to solve a system of linear equations where the transformation and result

matrices are large, i$.$e of dimension ~~$1001000$ in dimension. Submit your

code, plot$($s$)$ and comment on any issues you encountered doing this $($were there

any problems $($memory usage for instance$),$ how did you solve these?