Chapter 1, Problem 2E

Set $n=200$ and generate an $n\times n$ matrix and two vectors in ${ℝ}^n$ , both having integer entries, by setting
$\begin{array}{l}A=\text{floor}(10*\text{rand}(n));\\ b=\text{sum}(A\text{'})\text{'};\\ z=\text{ones}(n,1);\end{array}$
(Since the matrix and vectors are large, we use semicolons to suppress the printout.)
(a) The exact solution of the system $Ax=b$ should be the vector z. Why? Explain. One could compute the solution in MATLAB using the “\” operation or by computing $A^{-1}$ and then multiplying $A^{-1}$ times b. Let us compare these two computational methods for both speed and accuracy. One can use MATLAB’s $\text{tic}$ and $\text{toc}$ commands to measure the elapsed time for each computation. To do this, use the commands.
$\begin{array}{l}\text{tic},x=A\backslash b;\text{toc}\\ \text{tic},y=inv(A)*b;\text{toc}\end{array}$
Which method is faster?
To compare the accuracy of the two methods, we can measure how close the computed solutions x and y are to the exact solution z. Do this with the commands.
$\begin{array}{l}\text{max(abs}(\text{x-z}))\\ \text{max(abs}(\text{y-z}))\end{array}$
Which method produces the most accurate solution?
(b) Repeat part (a), using $n=500$ and $n=1000$ .