In this programming assignment, you will implement the Gaussian elimination algorithm to

solve systems of linear equations. Your task is to write a Python program that takes as input

a $k(k+1)$ matrix that represents a $kk$ system of linear equations. If there is a unique

solution, it should output that solution as a list of values for the corresponding variables. If

there is more than one solution, it should output one of those. Finally, if there are no

solutions for the system, your program should output the number $0.$

You should use an intermediate function that takes the augmented matrix of the system and

returns its reduced row$-$echelon form.

It is important to note that the proper implementation of the Gauss elimination algorithm is

crucial for this assignment, and you must explain how you have implemented the algorithm

in your program.

Python implementations

You are required to implement the following functions in your program:

reduced$\_$re$\_$form$($matrix$)$: This function takes an input a matrix $A$ of size $n(n+1).$ A

is the augmented matrix for a system of $n$ linear equations on $n$ variables. Its output

is the reduced row$-$echelon form of $A.$

solve$($matrix$)$: This function takes an input a matrix $A$ as above and the output is a

solution for the system, in the form of a list of values for the corresponding variables.

If the system is inconsistent, then solve$($A$)$ should return $0.$ If there are multiple

solutions, then any solution is accepted.

Constraints

Do not use any python libraries or external packages.

## template for the programming assignment $1$

## The function takes as input a matrix A of size n x $($n$+1)$

## A is the augmented matrix for a system of n linear equations on n variables

## the output is the $($reduced$)$ row$-$echelon form of A

## A matrix is given as a double array, eg:

## A $=[[3\mathrm{.}0,2\mathrm{.}0,-4\mathrm{.}0,3\mathrm{.}0],$

## $[2\mathrm{.}0,3\mathrm{.}0,3\mathrm{.}0,15\mathrm{.}0],$

## $[5\mathrm{.}0,-3,1\mathrm{.}0,14\mathrm{.}0]]$

def reduced$\_$re$\_$form$($A$)$:

n $=$ len$($A$)$

for i in range$($n$)$:

# bring the area of the matrix below and to the right of

# cell i$,$i to reduced row$-$echelon form

pass

return A

## a function that takes as input a matrix A of size n x $($n$+1)$

## A is the $($reduced$)$ row$-$echelon form of an augmented matrix

## for a system of n linear equations on n variables

## the output is a solution for the system, in the form of a

## list of values for the corresponding variables

def solve$\_$row$\_$echelon$($A$)$:

n $=$ len$($A$)$

x $=[0$ for $\_$ in range$($n$)]$

## fill in here

return x

## if the system is inconsistent, then solve$($A$)$ should return $0$

## if there are multiple solutions, then any will do

def solve$($A$)$:

n $=$ len$($A$)$

B $=$ reduced$\_$re$\_$form$($A$)$

return solve$\_$row$\_$echelon$($B$)$