# 05 In-Class Assignment: Gauss-Jordan¶

## Today’s Agenda (80 minutes)¶

```
#Load Useful Python Libraries
%matplotlib inline
import matplotlib.pylab as plt
import numpy as np
import sympy as sym
sym.init_printing(use_unicode=True)
```

```
---------------------------------------------------------------------------
ModuleNotFoundError Traceback (most recent call last)
<ipython-input-1-edd51b6ed677> in <module>
1 #Load Useful Python Libraries
----> 2 get_ipython().run_line_magic('matplotlib', 'inline')
3 import matplotlib.pylab as plt
4 import numpy as np
5 import sympy as sym
~/REPOS/MTH314_Textbook/MakeTextbook/envs/lib/python3.9/site-packages/IPython/core/interactiveshell.py in run_line_magic(self, magic_name, line, _stack_depth)
2342 kwargs['local_ns'] = self.get_local_scope(stack_depth)
2343 with self.builtin_trap:
-> 2344 result = fn(*args, **kwargs)
2345 return result
2346
~/REPOS/MTH314_Textbook/MakeTextbook/envs/lib/python3.9/site-packages/decorator.py in fun(*args, **kw)
230 if not kwsyntax:
231 args, kw = fix(args, kw, sig)
--> 232 return caller(func, *(extras + args), **kw)
233 fun.__name__ = func.__name__
234 fun.__doc__ = func.__doc__
~/REPOS/MTH314_Textbook/MakeTextbook/envs/lib/python3.9/site-packages/IPython/core/magic.py in <lambda>(f, *a, **k)
185 # but it's overkill for just that one bit of state.
186 def magic_deco(arg):
--> 187 call = lambda f, *a, **k: f(*a, **k)
188
189 if callable(arg):
~/REPOS/MTH314_Textbook/MakeTextbook/envs/lib/python3.9/site-packages/IPython/core/magics/pylab.py in matplotlib(self, line)
97 print("Available matplotlib backends: %s" % backends_list)
98 else:
---> 99 gui, backend = self.shell.enable_matplotlib(args.gui.lower() if isinstance(args.gui, str) else args.gui)
100 self._show_matplotlib_backend(args.gui, backend)
101
~/REPOS/MTH314_Textbook/MakeTextbook/envs/lib/python3.9/site-packages/IPython/core/interactiveshell.py in enable_matplotlib(self, gui)
3511 """
3512 from IPython.core import pylabtools as pt
-> 3513 gui, backend = pt.find_gui_and_backend(gui, self.pylab_gui_select)
3514
3515 if gui != 'inline':
~/REPOS/MTH314_Textbook/MakeTextbook/envs/lib/python3.9/site-packages/IPython/core/pylabtools.py in find_gui_and_backend(gui, gui_select)
278 """
279
--> 280 import matplotlib
281
282 if gui and gui != 'auto':
ModuleNotFoundError: No module named 'matplotlib'
```

## 2. Generalize the procedure¶

We are going to think about Gauss-Jordan as an algorithm. First I want you to think about how you would generalize the procedure to work on any matrix. Do the following before moving on to the next section.

✅**DO THIS**: Use the following matrix to think about how you would solve any system of equations using the Gauss-Jordan elimination algorithm. Focus on the steps.

✅**QUESTION**: What are the first three mathematical steps you would do to put the above equation into a reduced row echelon form using Gauss-Jordan method?

Put your answer here.

### Psudocode¶

✅**QUESTION**: Write down the steps you would complete to implement the Gauss-Jordan elimination algorithm as a computer programer. Some questions to answer:

What are the inputs?

What are the outputs?

How many and what types of loops would you have to guarantee success of your program?

Once you have thought this though the instructor will work with you to build the algorithm.

## 3. Basic Gauss Jordan¶

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:

✅**QUESTION**: Explain why doesn’t the provided `basic_gauss_jordan`

function work on the matrix \(B\)?

Put your answer to the above question here.

✅**QUESTION**: Describe how you could modify matrix \(B\) so that it would work with `basic_gauss_jordan`

AND still give the correct solution?

Put your answer to the above question here.

```
# Put your code here
```

Written by Dr. Dirk Colbry, Michigan State University

This work is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License.