Multiple Choice Questions
Linear Algebra: Eigenvalues and Eigenvectors
Topic: Eigenvalues and Eigenvectors
Grade: 11
Question 1:
Which of the following matrices has eigenvalues of 2 and 3?
A) [1 0; 0 1]
B) [2 0; 0 3]
C) [3 0; 0 2]
D) [2 1; 1 2]
Answer: B) [2 0; 0 3]
Explanation: The eigenvalues of a matrix can be found by solving the characteristic equation, det(A – λI) = 0, where A is the matrix, λ is the eigenvalue, and I is the identity matrix. For this question, the characteristic equation is det(A – 2I) = 0 and det(A – 3I) = 0. Only matrix B satisfies both equations. For example, let\’s consider the matrix A = [2 0; 0 3]. The eigenvalues of A are indeed 2 and 3.
Question 2:
Which of the following matrices is diagonalizable?
A) [1 2; 3 4]
B) [1 0; 0 2]
C) [1 1; 0 1]
D) [1 -1; 1 1]
Answer: B) [1 0; 0 2]
Explanation: A matrix is diagonalizable if it has n linearly independent eigenvectors, where n is the size of the matrix. In this case, matrix B has distinct eigenvalues of 1 and 2, and the eigenvectors are linearly independent. Therefore, matrix B is diagonalizable. For example, let\’s consider the matrix A = [1 0; 0 2]. The eigenvectors of A are [1 0] and [0 1], which are indeed linearly independent.
Question 3:
For a 3×3 matrix, how many eigenvalues can it have?
A) 0
B) 1
C) 2
D) 3
Answer: D) 3
Explanation: The number of eigenvalues of a matrix is equal to its size. Therefore, a 3×3 matrix can have 3 eigenvalues. For example, let\’s consider the matrix A = [1 0 0; 0 2 0; 0 0 3]. The eigenvalues of A are indeed 1, 2, and 3.
Question 4:
Which of the following statements about eigenvectors is true?
A) Eigenvectors can be any non-zero vector.
B) Eigenvectors are always orthogonal to each other.
C) Eigenvectors are always linearly independent.
D) Eigenvectors can only exist for square matrices.
Answer: C) Eigenvectors are always linearly independent.
Explanation: Eigenvectors are nonzero vectors that only change by a scalar factor when multiplied by a matrix. Eigenvectors corresponding to distinct eigenvalues are always linearly independent. For example, consider the matrix A = [2 0; 0 2]. The eigenvectors of A are [1 0] and [0 1], which are indeed linearly independent.
Question 5:
What is the determinant of a matrix with complex eigenvalues?
A) 0
B) 1
C) -1
D) It depends on the matrix.
Answer: D) It depends on the matrix.
Explanation: The determinant of a matrix is equal to the product of its eigenvalues. If the matrix has complex eigenvalues, the determinant can be any complex number depending on the specific values of the eigenvalues. For example, let\’s consider the matrix A = [1 2; -2 1]. The eigenvalues of A are 1 + 2i and 1 – 2i, and the determinant is (1 + 2i)(1 – 2i) = 5.
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