Can a 3x3 matrix have 4 eigenvalues?

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Homework Help Overview

The discussion revolves around the properties of eigenvalues in relation to a 3x3 matrix. Participants are exploring whether it is possible for such a matrix to have four eigenvalues, referencing the characteristic equation derived from the determinant.

Discussion Character

  • Conceptual clarification, Mathematical reasoning

Approaches and Questions Raised

  • Participants discuss the process of finding eigenvalues, noting that it involves calculating the determinant to form the characteristic equation. There is a focus on the degree of this equation and the implications for the number of eigenvalues.

Discussion Status

Some participants assert that a 3x3 matrix cannot have four eigenvalues, referencing the degree of the characteristic polynomial. There is an ongoing exploration of the reasoning behind this conclusion, with questions about the proof and implications of the degree of the polynomial.

Contextual Notes

Participants mention recent learning on the topic and express uncertainty about the proof regarding the number of solutions to a polynomial equation of degree three.

nicknaq
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Homework Statement



Prove or disprove the title of this thread.

Homework Equations


AX=(lamda)X


The Attempt at a Solution


I don't know where to start
 
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nicknaq said:

Homework Statement



Prove or disprove the title of this thread.

Homework Equations


AX=(lamda)X


The Attempt at a Solution


I don't know where to start

Start with the process you use to find the eigenvalues of a 3 x 3 matrix, which involves a determinant to get the characteristic equation for the matrix. What degree equation would you expect to get?
 
Mark44 said:
Start with the process you use to find the eigenvalues of a 3 x 3 matrix, which involves a determinant to get the characteristic equation for the matrix. What degree equation would you expect to get?

an equation of degree 3
 
So it's not possible for a 3 x 3 matrix to have four eigenvalues, right?
 
Mark44 said:
So it's not possible for a 3 x 3 matrix to have four eigenvalues, right?

right. Is there any proof that I can say for why an equation of degree 3 cannot have 4 solutions?

I guess it's obvious though.
 
no its not possible. I had completed this topic only today in my class and here is one interesting question.

how to solve for eigen values. I think we need to take determinant. of A -[tex]\lambda[/tex] I

so we will get [tex]\lambda[/tex] cube in the equation which obviously will give three values of [tex]\lambda[/tex]
 

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