Let $Ain{R}^{nn}$ be a non$-$singular matrix.

a$.$ Show how Gaussian elimination with partial pivoting can be used to solve the $k$ linear systems

$A{x}_1=t_1,A{x}_2=t_2,...,A{x}_k=t_k$

in $\frac{n^3}{3}+O(k{n}^2)$ flops. You must show precisely how the matrix components of the $PA=LU$

factorization are used in each stage of your algorithm, and justify the final operation count.

$($Note: You do not need to give details of the Gaussian elimination algorithm$$you may assume the

factorization is available.$)$

b$.$ For $k=n$ and a suitable choice for $t_1,t_2,$dots,$t_n,$ your algorithm in $($a$)$ can be used to compute $A^{-1}.$

Explain how this can be done.

c$.$ A possible scheme for solving $Ax=b$ is to first compute $A^{-1}$ using $($b$),$ and then compute $x=A^{-1}b.$

Is this scheme preferable to the Gaussian elimination algorithm as discussed in lecture? Give operation

counts to justify your answer.