Question $7.($Please see Appendix for mathematical details prior to beginning this question.$)$ Write a Python program, while using only the built$-$in Python commands or numpy library, that efficiently computes

$S_n(A)={\displaystyle \underset{k=0}{\overset{n}{}}}(I-A{)}^k,$

for integer $n0.$

Question $8.$ Consider the set up in Question $7.$

$($a$)$ Under what conditions on $Ain{R}^{mm}$ is $S_n(A)$ a good approximation to $A^{-1}in{R}^{mm},$ for large ninZ $?$

$($b$)$ Postulate an expression for the error of approximation and compute this value for both $($i$)$ a matrix that meets the conditions in part $($a$)$ and $($ii$)$ a matrix that does not meet the conditions in part $($a$).$

Question $7$ Appendix

Recall the geometric series

$\frac{1}{1-r}=\underset{n}{lim}{\displaystyle \underset{k=0}{\overset{n}{}}}{r}^k,$ for

In particular

$\frac{1}{1-r}-{\displaystyle \underset{k=0}{\overset{n}{}}}{r}^k={\displaystyle \underset{k=n+1}{\overset{}{}}}{r}^k$

$=\frac{r^{n+1}}{1-r}.$

Define $S_n(r)={\displaystyle \underset{k=0}{\overset{n}{}}}{r}^k$ and note that

$|S_n(r)-\frac{1}{1-r}|=\frac{|r{|}^{n+1}}{|1-r|},$

which suggests that $S_n(r)$ is a good approximation to $\frac{1}{1-r}.$