Quiz on Eigenvalues and Eigenvectors
Test your understanding of eigenvalues and eigenvectors with this comprehensive quiz.
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1. Which of the following best defines an eigenvalue?
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A scalar such that Av = λv for a nonzero vector v and square matrix A.
A vector that satisfies Av = v for any matrix A.
A matrix that commutes with every vector.
Other
2. Given the matrix A = [[2, 0], [0, 3]], what are its eigenvalues?
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2 and 3
0 and 1
3 and 5
Other
3. Select all statements that are true about eigenvectors.
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An eigenvector cannot be the zero vector.
Eigenvectors corresponding to different eigenvalues are always linearly independent.
Every square matrix has at least one eigenvector.
Other
4. Calculate the eigenvalues of the matrix A = [[4, 2], [1, 3]]. Enter your answers separated by a comma.
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5. For the matrix A = [[1, 2], [2, 1]], which of the following vectors is an eigenvector?
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[1, 1]
[1, -1]
[2, 2]
Other
6. Match each matrix with its correct set of eigenvalues.
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Rows
Eigenvalues
A = [[2,0],[0,3]]
{2,3}
{1,-1}
{0,0}
{5,5}
B = [[1,1],[1,1]]
{2,3}
{1,-1}
{0,0}
{5,5}
C = [[0,0],[0,0]]
{2,3}
{1,-1}
{0,0}
{5,5}
D = [[5,0],[0,5]]
{2,3}
{1,-1}
{0,0}
{5,5}
7. In your own words, explain what it means for a matrix to be diagonalizable.
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8. Rate your confidence in your answers to this quiz.
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9. For the matrix A = [[0, 1], [-2, -3]], find one eigenvector corresponding to the eigenvalue -1.
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10. Select the applications where eigenvalues and eigenvectors are commonly used.
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Principal Component Analysis (PCA)
Quantum Mechanics
Google's PageRank Algorithm
Other
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