Gaussian Elimination rules - apply to rows only?

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SUMMARY

The discussion centers on the application of Gaussian Elimination (GE) rules, specifically whether they apply only to rows or can also be applied to columns. The three established rules of GE include swapping rows, multiplying a row by a constant, and adding a multiple of one row to another. It is concluded that while these rules are primarily intended for rows, they can be applied to columns when calculating determinants, as transposing the matrix allows for such operations without altering the underlying equations.

PREREQUISITES
  • Understanding of Gaussian Elimination rules
  • Familiarity with matrix operations
  • Basic knowledge of determinants in linear algebra
  • Ability to solve systems of linear equations
NEXT STEPS
  • Study the properties of determinants in linear algebra
  • Learn about matrix transposition and its effects on operations
  • Explore advanced applications of Gaussian Elimination in solving linear systems
  • Investigate the implications of row operations on matrix rank and solutions
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Students of linear algebra, educators teaching matrix theory, and anyone seeking to deepen their understanding of Gaussian Elimination and its applications in solving equations and determinants.

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Homework Statement


In this video;

the uploader states at 0:47 that there are three rules.

1. Can swap rows
2. Can multiply a row by a constant
3. Can add a multiple of one row to another

Do these rules apply only to rows? Can they be applied to columns? The uploader has not stated this.

Thank you.



Homework Equations





The Attempt at a Solution

 
Last edited by a moderator:
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Think it depends on what you're trying to do via GE.

If you have 2 equations,

2x+3y=1
4x+2y=11

Then you will have a matrix

2 3
4 2

multiplied by

x
y

will be equal to

1
11

Finding solutions to this, it clearly makes no sense to multiple columns by a scalar, as this will change the equations, and the solutions.

However, if you're trying to find a determinant, then I think you can apply the same rules to columns, as you're allowed to transpose the matrix.
 
Thank you HmBe.
 

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