Can Column Operations be Used to Find the Inverse of a Matrix?

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Discussion Overview

The discussion revolves around the methods for finding the inverse of a matrix, specifically comparing the use of column operations to traditional row operations. Participants explore the implications of using column operations and whether they can achieve the same results as row operations in the context of matrix inversion.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested

Main Points Raised

  • One participant suggests augmenting a matrix with the identity matrix and using row operations to find its inverse, questioning if column operations would yield similar results.
  • Another participant asks for clarification on what constitutes column operations.
  • A participant asserts that the identity matrix should be adjoined below the original matrix rather than to its right.
  • There is a correction regarding terminology, with one participant clarifying that "RRE form" should be referred to as "reduced column echelon form."
  • One participant expresses uncertainty about the possibility of using both row and column operations simultaneously for finding an inverse.
  • Another participant proposes that column operations might be equivalent to row reducing the transpose of the matrix.
  • A later reply indicates that while column operations can be seen as related to row operations, they specifically manipulate the columns of the matrix.
  • It is mentioned that both row and column operations can be used algebraically to find an inverse, suggesting a relationship between the operations through matrix equations.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the effectiveness of column operations for finding the inverse of a matrix, and there are multiple competing views regarding the method and its implications.

Contextual Notes

There are limitations in the discussion regarding the specific bookkeeping methods for combining row and column operations, as well as the assumptions about the equivalence of operations on the transpose.

rock.freak667
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If I have a matrix

[tex] A= \left(<br /> \begin{array}{ccc}<br /> 1 & 2 & 3\\<br /> 0 & -1 & 4\\<br /> 1 & 1 & 6<br /> \end{array}<br /> \right)[/tex]
and I need to find [itex]A^{-1}[/itex] I would just augment with the identity matrix and then do row operations. But if I want to use column operations instead does it work in the same manner? because I think if use the column operations, the matrix A would be reduced to RRE form but nothing will happen to the identity matrix.
(Not too sure if I was clear about my problem.)
 
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what are column operations?
 
You need to adjoin an identity matrix below the original, not to its right.
 
rock.freak667 said:
I think if use the column operations, the matrix A would be reduced to RRE form
You mean reduced column echelon form.
 
Hurkyl said:
You mean reduced column echelon form.
Yeah sorry about that.That is what I meant.
Hurkyl said:
You need to adjoin an identity matrix below the original, not to its right.

Below it? So there is no way to both row and column operations at the same time to find the inverse or whatever?Also how would I solve a system of equations using column operations only?
 
wouldn't these column operations be the same thing as row reducing the transpose?
 
ice109 said:
wouldn't these column operations be the same thing as row reducing the transpose?

It is the same thing basically except that column operations uses the columns of the matrix,
 
rock.freak667 said:
Yeah sorry about that.That is what I meant.


Below it? So there is no way to both row and column operations at the same time to find the inverse or whatever?
Not with this particular bookkeeping method. Of course you can use both kinds of operations to find an inverse: algebraically, if R is the matrix denoting your row operations and C is the one denoting your column operations, then if you reduce your matrix to the identity, that says
RAC = I​
which you can easily solve for A.
 
oh okay then thanks
 

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