Linear Algebra: if the rank r of matrix A of order n

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• 1. The rank of matrix A is 1. It is proved that the eigenvalue of a has n-1 zeros? The rank of matrix A is 1. It is proved that the eigenvalue of a has n-1 zeros and another eigenvalue is the sum of diagonal elements Would you like to be more detailed? What does this diagram show?
• 2. If matrix a satisfies: AAT = E and | a | = - 1, then matrix a must have an eigenvalue of - 1. Why is it equal to proving that the determinant of | a + e | is 0
• 3. Let | a | = D ≠ 0 be the determinant of matrix A of order n, and find | a*| Adjoint matrix of a
• 4. Linear Algebra: there is a problem -- let a = (1,0, - 1) t, matrix A = AAT, n be a positive integer, find the determinant of ae-a ^ n? There is a sentence in the answer: from R (a) = 1, we know that the eigenvalues of a are 2,0,0?
• 5. Let the eigenvalues of the third order invertible matrix a be 1, 2 and 4 respectively, then [i-2a ^ - 1] is equal to It's the value of the determinant,
• 6. It is proved that the necessary and sufficient condition of invertibility of matrix A is that its eigenvalues are not equal to zero
• 7. How to prove that matrix A is invertible if and only if 0 is not the eigenvalue of a GOT IT
• 8. Is the invertibility of (a + b) matrix equal to the invertibility of a + if not, what
• 9. Let 2 be an eigenvalue of invertible matrix A, then an eigenvalue of 3A ^ 2 + e is
• 10. Let the sum of all elements of nonsingular matrix a be 2, then the matrix (1 / 3A ^ 2) ^ - 1 has an eigenvalue equal to () (a) 4 / 3; (b) 3 / 4;
• 11. In linear algebra, what is the relationship between the rank of a matrix and its eigenvalues?
• 12. Linear Algebra: let the eigenvalue of the real symmetric matrix A of order 3 be A1 = - 1, A2 = A3 = 1, and the eigenvector corresponding to A1 be B1 = (0,0,1) t, then calculate the matrix A
• 13. Linear algebra proves that the eigenvectors A1 and A2 corresponding to different eigenvalues of real symmetric matrix a must be orthogonal
• 14. Given the three eigenvalues A1 = 0, A2 = A3 = 2 of the real symmetric matrix A of order 3, and the eigenvector corresponding to the eigenvalue 0 is (1,0, - 1) ^ t, find the matrix A
• 15. Linear Algebra: why is the third order real symmetric matrix A, R (a-2e) = 1, so 2 is the double eigenvalue of a?
• 16. In linear algebra, if the rank of a matrix is increased by 1 or unchanged, how to prove it
• 17. A linear algebra: A is a matrix of order n, R (a) = R
• 18. Proof of rank of linear algebraic matrix and adjoint matrix At (3), I really don't understand why when R (a) & lt; n-1, all the N-1 order subformulas of a are 0? Why (3) step, R (a)
• 19. It is proved that there is an invertible matrix in a B, if a is invertible, then R (AB) = R (b) = R (BA)
• 20. Let a and B be invertible matrices of order n, and prove that (a *) * = | a | ^ n-2 · a