$Ax=b$

$A=$ np$.$random.normal $,$ size $=(n,n)$ np$.$identity$(n)$

$b=$ np$.$random.normal $,$ size $=(n,1)$

Two cases: $(1)n=150,(2)n=250($here $np$ means numpy$)$

$($i$)$ Show that the above linear equation has a unique solution

$($ii$)$ Use Python or MATLAB to construct the upper triangular and back substitution algorithm to find the solution

$($iii$)$ Use Python or MATLAB to construct the LU decomposition algoroithm to find the solution $($you may need the permutation step to do the LU decomposition$)$

$($iv$)$ Verify your solution using numpy.linalg.solve