# Can a 3x3 matrix have 4 eigenvalues?

• nicknaq
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.

## Homework Statement

Prove or disprove the title of this thread.

AX=(lamda)X

## The Attempt at a Solution

I don't know where to start

nicknaq said:

## Homework Statement

Prove or disprove the title of this thread.

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 -$$\lambda$$ I

so we will get $$\lambda$$ cube in the equation which obviously will give three values of $$\lambda$$

## 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.