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題組內容

109台大電信丙組

9.(18%) Let

whose eigenvalues are -5, -2 and l.

(a) (6%) Plcase find the eigenvalues of.

其他申論題

(b) What are the eigenvectors of this matrix?

(c) What is the determinant of this matrix?

(d) When will this matrix have its inverse? Please tell us your reason.

(e) Please evaluate (A - 2I)(A - 4I)(A - 51) + A, where I is a 3-by-3 identity matrix.

(b) (6%) Please find the eigenvalues of 2A.

(c) (6%) Please find the eigenvalues of A2.

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.

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.

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.

Let .Show that is linear combination of A and I2.

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