Linear Algebra: Inverse of Elementary Matrix

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Homework Help Overview

The discussion revolves around finding the inverse of a given 2x2 elementary matrix, specifically the matrix with elements 1, 0, 3, and -1. Participants are exploring the properties of elementary matrices and the process of calculating their inverses.

Discussion Character

  • Conceptual clarification, Mathematical reasoning, Problem interpretation

Approaches and Questions Raised

  • Participants discuss the proposed inverse of the matrix and question whether the sign of the constant term should be changed. There are inquiries about the relationship between the original matrix and the identity matrix when multiplied by the proposed inverse.

Discussion Status

The discussion is active, with participants providing different perspectives on the calculation of the inverse. Some guidance has been offered regarding the necessity of changing signs and the role of the determinant in the process. Multiple interpretations of the problem are being explored.

Contextual Notes

There is mention of Cramer’s rule and the determinant's significance in the context of finding the inverse, indicating that participants are considering various methods for solving the problem.

ephemeral1
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Homework Statement



What is the inverse of this elementary matrix? 2x2 matrix
1 0
3 -1


E=

Homework Equations



None

The Attempt at a Solution


This is what I think:
1 0
-3 -1

Is that right? Or do you leave the 3 positive? If so, please explain. Thank you.
 
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ephemeral1 said:

Homework Statement



What is the inverse of this elementary matrix? 2x2 matrix
1 0
3 -1


E=

Homework Equations



None

The Attempt at a Solution


This is what I think:
1 0
-3 -1

Is that right? Or do you leave the 3 positive? If so, please explain. Thank you.

Do you get the identity matrix when you multiply your inverse by the original?
 
No, I did not. But when I multiply
1 0
3 -1 by the original, I get the identity matrix. I don't understand why the 3 is positive. Don't we have to change the sign on additional constant? I thought the 3 is the additional constant? Or is it because the 1 on the second row is negative, we don't have to change the sign on the -3 when doing the inverse? Please explain. Thank you.
 
What method did you use to calculate the inverse? That should answer your question.
 
you are correct you must change the sign of 3, however the determinant is equal to -1. remember you have to divide by the determinant when using cramer's rule.
 
another way to calculate the inverse is to write the matrix equation AX=I and solve for the unknowns in X.
 

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