Consider the matrix

A $=$

$$

$$

$123$

$456$

$789$

$$

$,$ b $=$

$$

$$

$6$

$15$

$24$

$$

$$

Note that the $$true$$ solution i$.$e$.$ the x so that b $=$ Ax is x $=111$T $.$

You can assemble this matrix as follows:

import numpy as np

A $=(1+$ np$.$ arange $(9)).$ astype $($ float $).$ reshape $((3,3))$

$($a$)$ Solve this linear system by hand i$.$e$.$ find the general solution, and find a particular solution.

$($b$)$ Solve this linear system on a computer. Compute $,$ condition number estimate, and up$-$

per bound for the error. What do you observe? Depending on where you do this

calculation, you will not get what I expected. For the time being skip this

part, and continue on to $($c$).$

$($c$)$ Now replace the matrix by

A $=$

$$

$$

$123$

$45+10156$

$789+1015$

$$

$.$

What do you observe now?

$($d$)($optional$)$ What happens as you vary the exponent of the perturbation i$.$e$.10$p for

p in $[10,16].$ Consider integer steps. Explain what you observe