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11.4: Elementary Matrices

Last updated

Sep 17, 2022
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- 11.3: Identity Matrix
- 11.5: Solving Many Systems (at the same time)

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from urllib.request import urlretrieve
urlretrieve('https://raw.githubusercontent.com/colbrydi/jupytercheck/master/answercheck.py', 
            'answercheck.py');

from urllib.request import urlretrieve
urlretrieve('https://raw.githubusercontent.com/colbrydi/jupytercheck/master/answercheck.py', 
            'answercheck.py');

NOTE: A detailed description of elementary matrices can be found here in the Beezer text Subsection EM 340-345 if you find the following confusing.

There exist a cool set of matrices that can be used to implement Elementary Row Operations. Recall our elementary row operations include:

Swap two rows
Multiply a row by a constant ( $c$ )
Multiply a row by a constant ( $c$ ) and add it to another row.

You can create these elementary matrices by applying the desired elementary row operations to the identity matrix.

If you multiply your matrix from the left using the elementary matrix, you will get the desired operation.

For example, here is the elementary row operation to swap the first and second rows of a $3 \times 3$ matrix:

$\begin{split} E_{12}= \left[ \begin{matrix} 0 & 1 & 0\\ 1 & 0 & 0 \\ 0 & 0 & 1 \end{matrix} \right] \end{split} \nonumber$

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import numpy as np
import sympy as sym
sym.init_printing(use_unicode=True)
A = np.matrix([[3, -3,9], [2, -2, 7], [-1, 2, -4]])
sym.Matrix(A)
# 'Run' this cell to see the output

import numpy as np
import sympy as sym
sym.init_printing(use_unicode=True)
A = np.matrix([[3, -3,9], [2, -2, 7], [-1, 2, -4]])
sym.Matrix(A)
# 'Run' this cell to see the output

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E1 = np.matrix([[0,1,0], [1,0,0], [0,0,1]])
sym.Matrix(E1)
# 'Run' this cell to see the output

E1 = np.matrix([[0,1,0], [1,0,0], [0,0,1]])
sym.Matrix(E1)
# 'Run' this cell to see the output

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A1 = E1*A
sym.Matrix(A1)
# 'Run' this cell to see the output

A1 = E1*A
sym.Matrix(A1)
# 'Run' this cell to see the output

Do This

Give a $3 \times 3$ elementary matrix named E2 that swaps row 3 with row 1 and apply it to the $A$ Matrix. Replace the matrix $A$ with the new matrix.

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# Put your answer here.

# Put your answer here.

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from answercheck import checkanswer

checkanswer.matrix(E2,'2c2d2e407389eabeb6d90894565c830f');

from answercheck import checkanswer

checkanswer.matrix(E2,'2c2d2e407389eabeb6d90894565c830f');

Do This

Give a $3 \times 3$ elementary matrix named E3 that multiplies the first row by $c=3$ and adds it to the third row. Apply the elementary matrix to the $A$ matrix. Replace the matrix $A$ with the new matrix.

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# Put your answer here.

# Put your answer here.

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from answercheck import checkanswer

checkanswer.matrix(E3,'55ae1f9eb21df00c59dad623b9471506');

from answercheck import checkanswer

checkanswer.matrix(E3,'55ae1f9eb21df00c59dad623b9471506');

Do This

Give a $3 \times 3$ elementary matrix named E4 that multiplies the second row by a constant $c=1/2$ applies this to matrix $A$ .

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# Put your answer here.

# Put your answer here.

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from answercheck import checkanswer

checkanswer.matrix(E4,'3a5256840ef907a1b73ebba4471ac26d');

from answercheck import checkanswer

checkanswer.matrix(E4,'3a5256840ef907a1b73ebba4471ac26d');

If the above are correct then we can combine the three operators on the original matrix $A$ as follows.

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A = np.matrix([[3, -3,9], [2, -2, 7], [-1, 2, -4]])

sym.Matrix(E4*E3*E2*A)

A = np.matrix([[3, -3,9], [2, -2, 7], [-1, 2, -4]])

sym.Matrix(E4*E3*E2*A)

Do This

Do This

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