10.3: Basic Gauss Jordan

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#  Load Useful Python Libraries
import numpy as np
import sympy as sym
sym.init_printing(use_unicode=True)

The following is implementation of the Basic Gauss-Jordan Elimination Algorithm for Matrix \(A^{m \times n}\) (Pseudocode):

for i from 1 to m:
    for j from 1 to m    
        if i ≠ j:
            Ratio = A[j,i]/A[i,i]
            #Elementary Row Operation 3
            for k from 1 to n:
                A[j,k] = A[j,k] - Ratio * A[i,k]
            next k
        endif
    next j
    
    #Elementary Row Operation 2
    Const = A[i,i]
    for k from 1 to n:
        A[i,k] = A[i,k]/Const
next i

Do This

using the Pseudocode provided above, write a basic_gauss_jordan function which takes a list of lists \(A\) as input and returns the modified list of lists:

# Put your answer here.

Lets check your function by applying the basic_gauss_jordan function and check to see if it matches the answer from matrix \(A\) in the pre-class video:

A = [[1, 1, 1, 2], [2, 3, 1, 3], [0, -2, -3, -8]]
answer = basic_gauss_jordan(A)
sym.Matrix(answer)

answer_from_video = [[1, 0, 0, -1], [0, 1, 0, 1], [0, 0, 1, 2]]
np.allclose(answer, answer_from_video)

The above psuedocode does not quite work properly for all matrices. For example, consider the following augmented matrix:

\[\begin{split}
B = \left[
\begin{matrix}
0 & 1 & 33\\
5 & 3 & 7 \\
6 & 69 & 4
\end{matrix}
\, \middle\vert \,
\begin{matrix}
30 \\
90 \\
420
\end{matrix}
\right]
\end{split} \nonumber \]

Question

Explain why doesn’t the provided basic_gauss_jordan function work on the matrix \(B\)?

Question

Describe how you could modify matrix \(B\) so that it would work with basic_gauss_jordan AND still give the correct solution?