6 8. Let A be a square matrix one of whose eigenvalues are 1. Is 12...
4. (a) (6 marks) Let A be a square matrix with eigenvector v, and corresponding eigenvalue 1. Let c be a scalar. Show that A-ch has eigenvector v, and corresponding eigenvalue X-c. (b) (8 marks) Let A = (33) i. Find the eigenvalues of A. ii. For one of the eigenvalues you have found, calculate the corresponding eigenvector. iii. Make use of part (a) to determine an eigenvalue and a corresponding eigenvector 2 2 of 5 - 1
Let matrix M = -8 -24 -12 0 4 0 6 12 10 (a) Find the eigenvalues of M (b) For each eigenvalue λ of M, find a basis for the eigenspace of λ. (c) Is the matrix M diagonalizable? If so, find matrices D and P such that D is a diagonal matrix and M=PDP^−1. If not, explain carefully why not.
Let matrix M = -8 -24 12 0 4 0 6 12 10 (a) Find the eigenvalues of M (b) For each eigenvalue λ of M, find a basis for the eigenspace of λ. (c) Is the matrix M diagonalizable? If so, find matrices D and P such that D is a diagonal matrix and M=PDP−1. If not, explain carefully why not.
(12) (7 marks) Let the distinct eigenvalues of a square matrix A be denoted by 11, ..., dk. Suppose the corresponding algebraic multiplicities are m1, ..., mk and that A is similar to an upper-triangular matrix. Show that k tr(A) = midi and det(A) = II (1;)" i=1 i=1
6) Let A be a 5x5 matrix, with 3 different eigenvalues, and let a be an eigenvalue of multiplicity 3. If A-21 has rank 2, is A defective? (Explain, as a yes or no answer will receive no credit).
-8 -24 -12 (16 points) Let A= 0 4 0 6 12 10 (a) (4 points) Find the eigenvalues of A. (b) [6 points) For each eigenvalue of A, find a basis for the eigenspace of (b) [6 points) is the matrix A diagonalizable? If so, find matrices D and P such that is a diagonal matrix and A = PDP 1. If not, explain carefully why not.
66. Suppose a non-homogeneous system AF = 5 of six linear equations in eight variables has a solution, with two free variablea. Is is possible that Až = is inconsistent for some y in R6? Why or why not? 67. Let A be a 4 x 4 matrix. The eigenvectors of A are 6 and - 5. The eigenspace corresponding to 1 = 6 is 2-dimensional and the eigenspace corresponding to A = -5 is 1-dimensional. Is A diagonalizable? Why...
6) Let A be a 5x5 matrix, with 3 different eigenvalues, and let 2 be an eigenvalue of multiplicity 3. If A-2/ has rank 2, is A defective? (Explain, as a yes or no answer will receive no credit).
6) Let A be a 5x5 matrix, with 3 different eigenvalues, and let i be an eigenvalue of multiplicity 3. If A-11 has rank 2, is A defective? (Explain, as a yes or no answer will receive no credit).
(1 point) The matrix 4-4 A 0 -8 0 4 has two real eigenvalues, one of multiplicity 1 and one of multiplicity 2. Find the eigenvalues and a basis for each eigenspace The eigenvalue A, is and a basis for its associated eigenspace is The eigenvalue A2 is and a basis for its associated eigenspace is