Let a be a square matrix of order n, AA = a, and prove that R (a) + R (A-E) = n

RELATED INFORMATIONS

• 1. On "let square matrix a satisfy a ^ 2-a-2e = 0, prove that a and a + 2E are invertible, and find the inverse matrix of a and the inverse matrix of (a + 2e)" I can't see a big mistake when a teacher or a classmate does this, A ^ 2-e = a + e, left square difference formula, The results are as follows (A+E)(A-E)=A+E, The two sides are multiplied by the inverse of (a + e) to get a = 2E, So the inverse of (a + e) is equal to E / 3 There is another way From the known equation A(A-E) = 2E So a [(1 / 2) (A-E)] = e So a is reversible and a ^ - 1 = (1 / 2) (A-E)
• 2. Let square matrix a satisfy a ^ 2-2a + 4E = O, prove that a + E and a-3e are invertible, and find their inverse matrix I know how to do it, but I don't understand why. Could you please explain how to do it? That is, list the process and explain each step. Thank you!
• 3. Let n-order matrix a satisfy a ^ 2 + 2A – 3E = 0, prove that a + 4E is invertible, and find their inverse
• 4. Given that the eigenvalues of matrix A of order 3 are 1,2,3, try to find the eigenvalues of B = 1 / 2A * + 3E
• 5. If a is a square matrix of order n, e is a unit matrix of order n, and a ^ 3 = O, it is proved that A-E is an invertible matrix!
• 6. Let n-order square matrix a satisfy a * A-A + e = 0, and prove that a is an invertible matrix
• 7. Let n-order real square matrix A = a ^ 2 and E be n-order identity matrix. It is proved that R (a) + R (A-E) = n
• 8. If we know that the fourth-order square matrix A is similar to B, and the eigenvalues of a are 2,3,4,5, then | B-I | =? (where I is the fourth-order identity matrix) Why is the eigenvalue of B-I 2-1,3-1,4-1,5-1, that is: 1,2,3,4
• 9. Let the eigenvalues of a square matrix of order 3 be 1,2,0, and its corresponding eigenvectors A1, A2, A3. B = a ^ 3-2a + 3E, and find the eigenvector of B ^ - 1 Why A1, A2, A3? It's not mentioned in the book. What's the basis
• 10. A. B is a square matrix of order n. It is proved that AB and Ba have the same eigenvalues
• 11. Let a be a square matrix of order n, and satisfy AA ^ t = E and | a | = - 1, prove that the determinant | e + a | = 0 My question is why |A| |E+A'| = |A| |(E+A)'| = |A| |E+A|
• 12. Let n square matrix a satisfy a ^ 2 = A and E be the identity matrix of order n, and prove that R (a) + R (A-E) = n
• 13. Let a be a 3-order square matrix with eigenvalues of 1,2, - 3, find the eigenvalues of a ^ 2-3a + A ^ - 1 + 2E, and | a ^ 2-3a + A ^ - 1 + 2e |, hope to give the process
• 14. Proof: let n-order square matrix a satisfy a ^ 2 = a, and prove that the eigenvalue of a is 1 or 0
• 15. Let the eigenvalues of the third-order square matrix a be - 1, - 2, - 3, and find a *, a & # 178; + 3A + E
• 16. Let the eigenvalues of the fourth-order square matrix a be 1 / 2,1 / 3,1 / 4,1 / 5, then | a ^ - 1-e | =?
• 17. It is known that the fourth-order square matrix a satisfies | A-E | = 0, square matrix B = a ^ 3-3a ^ 2, BB ^ t = 2E, and | B|
• 18. Let a be a matrix of order 4 × 3, C = AAT, then | C|=
• 19. Given that a is a matrix of order n and satisfies the equation A2 + 2A = 0, it is proved that the eigenvalue of a can only be 0 or - 2
• 20. Given that an eigenvalue of matrix A = 1a23 is - 1, find another eigenvalue of matrix A and an eigenvector belonging to λ