Can a 3x3 matrix have 4 eigenvalues?

In summary, the conversation discusses proving or disproving a title related to finding eigenvalues of a 3 x 3 matrix. The process of finding eigenvalues is mentioned, including using a determinant to get the characteristic equation which would result in an equation of degree 3. The impossibility of a 3 x 3 matrix having four eigenvalues is also mentioned, and the use of determinants to find eigenvalues is further discussed.
  • #1
67
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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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  • #2
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?
 
  • #3
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
 
  • #4
So it's not possible for a 3 x 3 matrix to have four eigenvalues, right?
 
  • #5
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.
 
  • #6
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]
 

1. Can a 3x3 matrix have 4 eigenvalues?

No, a 3x3 matrix can only have up to 3 distinct eigenvalues.

2. Is it possible for a 3x3 matrix to have complex eigenvalues?

Yes, a 3x3 matrix can have complex eigenvalues, as long as it has at least one complex entry in its matrix elements.

3. How can I find the eigenvalues of a 3x3 matrix?

You can find the eigenvalues of a 3x3 matrix by solving the characteristic equation, det(A-λI) = 0, where A is the matrix and λ is the variable representing the eigenvalue.

4. Can a 3x3 matrix have repeated eigenvalues?

Yes, a 3x3 matrix can have repeated eigenvalues, as long as the matrix is not diagonalizable. In this case, the matrix will have fewer than 3 distinct eigenvalues.

5. How many eigenvectors can a 3x3 matrix have?

A 3x3 matrix can have up to 3 linearly independent eigenvectors, corresponding to each of its eigenvalues.

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