Let a be a square matrix of order 2, a, β be linearly independent 2-dimensional sequence vectors, AA = 0, a β = a + β, then the nonzero eigenvalues of A

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• 1. If the 3-dimensional column vectors α and β satisfy α 'β = 2, how to find the nonzero eigenvalues of the matrix β α'?
• 2. Let a be a matrix of order n, | a | ≠ 0, a * be the adjoint matrix of a, and E be the unit matrix of order n. if a has an eigenvalue λ, then (a *) 2 + e must have an eigenvalue______ ．
• 3. Given that the matrix A of order n satisfies a ^ 2-2a-3e = 0, it is proved that the eigenvalue of a can only be - 1 or 3. How to prove that it can only be - 1? (- e-A) (3e-a) = 0, but how can we prove that it can only be - 1 or 3?
• 4. Let a matrix of order n have n eigenvalues 0,1,2,..., n-1, and B ~ A, and find det (I + b)
• 5. If the eigenvalue of matrix A is t, why is the eigenvalue of K power of a K power of T,
• 6. A is a matrix of order n. It is proved that tr (a ^ k) = the sum of the k-th power of the eigenvalues of A
• 7. Let a matrix of order 4 satisfy | 3e-a |, AAT = 2E, | a|
• 8. The eigenvalues of adjoint matrix A * are 1,2,4,8. Find the eigenvalues of (1 / 3a) ^ - 1
• 9. Let a be a square matrix of order n, 2,4,..., 2n be the n eigenvalues of a, and calculate the value of determinant / a-3e /
• 10. Let a be a square matrix of third order, and it is known that a has two eigenvalues - 1. - 2, and the rank of (a + 3e) is 2. Find the determinant of a + 4E
• 11. If the 3-dimensional column vectors α and β satisfy α t β = 2, then the nonzero eigenvalue of the matrix β α t is? Although the characteristic polynomials and eigenvalues of AB and Ba are the same, why is the nonzero eigenvalue equal to 2? To Yin Yang double edged sword: β α t β = β (α t β) = 2 β or not
• 12. A third-order matrix has only two linearly independent eigenvectors, and this matrix has only one eigenvalue of the triple root
• 13. Let a be a matrix of order 2, α 1 and α 2 be two linearly independent two-dimensional vectors, a α 1 = O, a α 2 = 2 α 1 + α 2, and find the non-zero eigenvalues of A
• 14. Let a be a matrix of order n and satisfy AAT = e. the determinant of a is less than zero. It is proved that - 1 is an eigenvalue of A
• 15. It is proved that if a is a matrix of order s * n, then the eigenvalues of ATA are all nonnegative real numbers
• 16. A is a nonzero matrix of order n. given a ^ 2 + a = 0, can we deduce that - 1 is an eigenvalue of a? Yes A^2+A=0 So f (x) = x ^ 2 + X is a zeroing polynomial of matrix A, The eigenvalue of a can only be the root of the zeroing polynomial f (x), that is, 0 or - 1, Because a is a nonzero matrix, it is impossible to have all zero eigenvalues, So there must be an eigenvalue - 1 A is a non-zero matrix, so when the eigenvalues are impossible, they are all zero? For example, a = [0 01; 0 00; 0 00]
• 17. Let all the elements of the fourth-order matrix a be 1, then the nonzero eigenvalue of a is 1
• 18. Let a and B be m * N and N * m matrices respectively. It is proved that AB and Ba have the same nonzero eigenvalues
• 19. A is a real symmetric matrix of third order, a ^ 2 + 2A = 0, R (a) = 2. Find all eigenvalues of a and the value of determinant | a ^ 2 + 3E | Why R (a) = 2, then - 2 is a double root?
• 20. It is known that the eigenvalues of a matrix of order 3 are 2,1, - 1. Find the eigenvalues of a + 3E and calculate the determinant | a + 3E|