$(3)$ Let P$=[2-26,131,402]$

$($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 $100->1000$ 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?