Linear Algebra : Eigenvalues and Eigenvectors

Study concepts, example questions & explanations for Linear Algebra

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Example Questions

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Example Question #1 : Eigenvalues And Eigenvectors

Find the Eigen Values for Matrix .

 

Possible Answers:

 

 

 

There are no Eigen Values

 

 

Correct answer:

 

Explanation:

The first step into solving for eigenvalues, is adding in a  along the main diagonal. 

Now the next step to take the determinant.

 

Now lets FOIL, and solve for .

Now lets use the quadratic equation to solve for .

 

So our eigen values are

 

Example Question #2 : Eigenvalues And Eigenvectors

Find the eigenvalues for the matrix

Possible Answers:

Correct answer:

Explanation:

The eigenvalues, ,  for the matrix are values for which the determinant of is equal to zero. First, find the determinant:

Now set the determinant equal to zero and solve this quadratic:

this can be factored:

The eigenvalues are 5 and 1.

 

Example Question #3 : Eigenvalues And Eigenvectors

Which is an eigenvector for , or

Possible Answers:

neither one is an Eigenvector

both and

Correct answer:

Explanation:

To determine if something is an eignevector, multiply times A:

Since this is equivalent to , is an eigenvector (and 5 is an eigenvalue).

This cannot be re-written as times a scalar, so this is not an eigenvector.

Example Question #4 : Eigenvalues And Eigenvectors

Find the eigenvalues for the matrix

Possible Answers:

Correct answer:

Explanation:

The eigenvalues are scalar quantities, , where the determinant of is equal to zero.

First, find an expression for the determinant:

Now set this equal to zero, and solve:

this can be factored (or solved in another way)

The eigenvalues are -5 and 3.

Example Question #5 : Eigenvalues And Eigenvectors

Which is an eigenvector for , or ?

Possible Answers:

Both and

Neither is an eigenvector

Correct answer:

Both and

Explanation:

To determine if something is an eigenvector, multiply by the matrix A: 

This is equivalent to so this is an eigenvector.

This is equivalent to so this is also an eigenvector.

 

Example Question #6 : Eigenvalues And Eigenvectors

Determine the eigenvalues for the matrix

Possible Answers:

Correct answer:

Explanation:

The eigenvalues are scalar quantities where the determinant of is equal to zero. First, write an expression for the determinant:

 this can be solved by factoring:

The solutions are -2 and -7

Example Question #7 : Eigenvalues And Eigenvectors

Which is an eigenvector for the matrix or

Possible Answers:

Both and

Neither one is an eigenvector

Correct answer:

Explanation:

To determine if a vector is an eigenvector, multiply with A:

. This cannot be expressed as an integer times , so is not an eigenvector

 

This can be expressed as , so is an eigenvector.

Example Question #8 : Eigenvalues And Eigenvectors

Possible Answers:

Correct answer:

Explanation:

Example Question #9 : Eigenvalues And Eigenvectors

Possible Answers:

Correct answer:

Explanation:

Example Question #10 : Eigenvalues And Eigenvectors

Possible Answers:

Correct answer:

Explanation:

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