$3.$ Let p$=[(2$&$-2$&$6$@$1$&$3$&$1$@$4$&$0$&$2)]$

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!

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!

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$?$

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?