Problem 1PP: Compute A8, where A = [4321]. Problem 2PP: Let A = [31227], v1 = [31], and v2 = [21]. Suppose you are told that v1 and v2 are eigenvectors of... Problem 3PP: Let A be a 4 4 matrix with eigenvalues 5, 3, and 2, and suppose you know that the eigenspace for =... Problem 1E: In Exercises 1 and 2, let A = PDP1 and compute A4. 1. P = [5723], D = [2001] Problem 2E: In Exercises 1 and 2, let A = PDP1 and compute A4. 2. P = [2335], D = [1001/2] Problem 3E: In Exercises 3 and 4, use the factorization A = PDP1 to compute Ak, where k represents an arbitrary... Problem 4E Problem 5E: In Exercises 5 and 6. the matrix A is factored in the form PDP1. Use the Diagonalization Theorem to... Problem 6E Problem 7E: Diagonalize the matrices in Exercises 720, if possible. The eigenvalues for Exercises 1116 are as... Problem 8E: Diagonalize the matrices in Exercises 720, if possible. The eigenvalues for Exercises 1116 are as... Problem 9E: Diagonalize the matrices in Exercises 720, if possible. The eigenvalues for Exercises 1116 are as... Problem 10E: Diagonalize the matrices in Exercises 720, if possible. The eigenvalues for Exercises 1116 are as... Problem 11E: Diagonalize the matrices in Exercises 720, if possible. The eigenvalues for Exercises 1116 are as... Problem 12E Problem 13E: Diagonalize the matrices in Exercises 720, if possible. The eigenvalues for Exercises 1116 are as... Problem 14E Problem 15E: Diagonalize the matrices in Exercises 720, if possible. The eigenvalues for Exercises 1116 are as... Problem 16E: Diagonalize the matrices in Exercises 720, if possible. The eigenvalues for Exercises 1116 are as... Problem 17E: Diagonalize the matrices in Exercises 720, if possible. The eigenvalues for Exercises 1116 are as... Problem 18E: Diagonalize the matrices in Exercises 720, if possible. The eigenvalues for Exercises 1116 are as... Problem 19E: Diagonalize the matrices in Exercises 720, if possible. The eigenvalues for Exercises 1116 are as... Problem 20E Problem 21E Problem 22E Problem 23E Problem 24E Problem 25E Problem 26E Problem 27E Problem 28E Problem 29E: A is a 5 5 matrix with two eigenvalues. One eigenspace is three-dimensional, and the other... Problem 30E: A is a 3 3 matrix with two eigenvalues. Each eigenspace is one-dimensional. Is A diagonalizable?... Problem 31E: A is a 4 4 matrix with three eigenvalues. One eigenspace is one-dimensional, and one of the other... Problem 32E: A is a 7 7 matrix with three eigenvalues. One eigenspace is two-dimensional, and one of the other... Problem 33E: Show that if A is both diagonalizable and invertible, then so is A1. Problem 34E: Show that if A has n linearly independent eigenvectors, then so does AT. [Hint: Use the... Problem 35E: A factorization A = PDP1 is not unique. Demonstrate this for the matrix A in Example 2. With D1 =... Problem 36E: With A and D as in Example 2, find an invertible P2 unequal to the P in Example 2, such that A =... Problem 37E: Construct a nonzero 2 2 matrix that is invertible but not diagonalizable. Problem 38E: Construct a nondiagonal 2 2 matrix that is diagonalizable but not invertible. Problem 39E Problem 40E Problem 41E Problem 42E format_list_bulleted