# Eigenvalue and Eigenvector in Linear Algebra Chapter Exam

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#### Question 1 1. How many eigenvalues does a 2 by 2 matrix have?

#### Question 2 2. Find the eigenvalues of the following matrix.

#### Question 3 3. How many eigenvectors does a 2 by 2 matrix have?

#### Question 4 4. One of the eigenvalues of the following matrix is -2. Find the corresponding eigenvector where we put 1 in for the first parameter of the vector.

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Question 5
5.
For any real number *x*, the following is an eigenvector of a matrix *A*. Find its unit eigenvector.

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Question 6
6.
To diagonalize a matrix, A, we calculate Pt AP. For the normalized eigenvectors, *u*1 and *u*2, the matrix, P, is given by _____.

#### Question 7 7. The matrix, A, is a _____.

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Question 8
8.
If the eigenvector of a matrix is the vector, *v*1, shown below, then the length of *v*1 is _____.

#### Question 9 9. To diagonalize a symmetric matrix, A, we calculate Pt AP. The result is the matrix, D, where D is the diagonal matrix shown. The eigenvalues of A are _____.

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Question 10
10.
For the eigenvector, *v*, the normalized eigenvector, *u*, is _____.

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#### Question 11 11. Find the eigenvectors associated with the eigenvalue of 4 for the following matrix.

#### Question 12 12. In order to find a matrix's eigenvectors, we solve the following equation. Which of this equation's variables represents the eigenvectors?

#### Question 13 13. Find the eigenvectors associated with the eigenvalue of 4 for the following matrix.

#### Question 14 14. Find the eigenvectors associated with the eigenvalue of 0 for the following matrix.

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Question 15
15.
Which of the following statements is **NOT** true?

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Question 16
16.
Find all the eigenvalues for the following 2x2 matrix (**A**).

#### Question 17 17. Let's say you work a problem where you find all the eigenvalues of a 5x5 matrix. How many eigenvalues will you find?

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Question 18
18.
What is the general equation we use to find the eigenvalues of any *nxn* matrix (A)?

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Question 19
19.
Find all the eigenvalues of the following 2x2 matrix (**A**).

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Question 20
20.
Find all the eigenvalues for the following 3x3 matrix (**A**).

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#### Question 21 21. The values in a diagonal form of a matrix are the _____ of the matrix.

#### Question 22 22. Which of the following matrices are in diagonal form?

#### Question 23 23. What determined the degree of the characteristic polynomial?

#### Question 24 24. In order for a matrix to have a diagonal form, we must be able to consider values for which the characterstic polynomial is equal to

#### Question 25 25. What sort of matrices may be diagonalized.

#### Eigenvalue and Eigenvector in Linear Algebra Chapter Exam Instructions

Choose your answers to the questions and click 'Next' to see the next set of questions. You can skip questions if you would like and come back to them later with the yellow "Go To First Skipped Question" button. When you have completed the practice exam, a green submit button will appear. Click it to see your results. Good luck!