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無年度 - 主題課程_對角化：特徵多項式和行列式#107845
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題組內容
Given the following matrix
(e) Please evaluate (A - 2I)(A - 4I)(A - 51) + A, where I is a 3-by-3 identity matrix.
其他申論題
(a) What are the eigenvalues of this matrix?
#462086
(b) What are the eigenvectors of this matrix?
#462087
(c) What is the determinant of this matrix?
#462088
(d) When will this matrix have its inverse? Please tell us your reason.
#462089
(a) (6%) Plcase find the eigenvalues of.
#462091
(b) (6%) Please find the eigenvalues of 2A.
#462092
(c) (6%) Please find the eigenvalues of A2.
#462093
109中興電機 (15 pts) The eigenvalues of A and AT are the same, because det(A- λ l) -det(A- λ l)T= det(AT- λ l). By coming up with a 2 X 2 counter-example, show an example that the cigenvectors of A and ATneed not be the same.
#462094
106中興電機 (10pts) Suppose A is a real and symmetric matrix with order n, and has a repeated eigenvalue. Then for every i in {1,2, ... n}, there exists an eigenvector whose ith component is 0.
#462095
105中興電機 (10pts) Suppose A and B are two matrices with size m✕n and n✕m resbectively. Show that the nonzero eigenvalues of AB and BA are the same.
#462096
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