Which of the following is/are the eigenvector(s) for the matrix given below?
2022
Which of the following is/are the eigenvector(s) for the matrix given below?
Attempted by 85 students.
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Method: To test whether a vector v is an eigenvector of matrix A, compute A·v and check whether the result is a scalar multiple of v.
For v = [-1, 1, 0, 1]^T: A·v = [-1, 1, 0, 1]^T = 1·v, so this is an eigenvector with eigenvalue 1.
For v = [1, 0, -1, 0]^T: A·v = [-7, -5, 12, 25]^T, which is not a scalar multiple of [1, 0, -1, 0]^T, so this is not an eigenvector.
For v = [-1, 0, 2, 2]^T: A·v = [-3, 0, 6, 6]^T = 3·v, so this is an eigenvector with eigenvalue 3.
For v = [0, 1, -3, 0]^T: A·v = [0, 3, -9, 0]^T = 3·v, so this is an eigenvector with eigenvalue 3.
Final answer: The eigenvectors among the given choices are the vectors [-1, 1, 0, 1]^T (eigenvalue 1), [-1, 0, 2, 2]^T (eigenvalue 3), and [0, 1, -3, 0]^T (eigenvalue 3). The vector [1, 0, -1, 0]^T is not an eigenvector.