Consider the following matrix to the right, where a and b are real numbers: [1 a b 2]
A) Calculate its eigenvalues
B) For a = 1, b = 0 calculate its eigenvector
Consider the following matrix to the right, where a and b are real numbers: [1 a...
Consider the matrix A where a, b, c are real numbers. In this problem, we consider what the ca cb dimensions of subspaces associated to A could be; these answers may depend on the values of a, b, and c, and we want to see what the possible dimensions are. (a) Could the null space of A be 0-dimensional? 1-dimensional? 2-dimensional? (b) Could the column space of A be 0-dimensional? 1-dimensional? 2-dimensional?
5. The following matrix B has known eigenvalues λ1-1 and λ2-6. 10a-1 B-0b-23 c30 0 Where a, b and c real numbers and vis the eigenvector associated with the eigenvalue A1. e. Determine as many of a, b, and c as you can. f.Determine the third eigenvalue, if possible. g.Determine the second and third eigenvectors, if possible.
(b) The matrix B= 1 2 2 3 3 1 3 2 4 has eigenvalues 7,2, -1. i. Find a column and a row eigenvector of B corresponding to the Perron eigenvalue. ii. Find a rank one nonnegative matrix C such that the matrix B+C will have eigenvalues 13, 2, -1. iii. Let a and B be real numbers. Calculate the eigenvalues of D(a, b) = aB+ BC. iv. Find limno(+B)"
How to do Part 3? -- Find e^(At), the exponential of matrix A, where t ∈ ℝ is any real number. Part 1: Finding Eigenpairs [10 10 5 10 -5 Find the eigenvalues λ,A2 and their corresponding eigenvectors vi , v2 of the matrix A- (a) Eigenvalues: 1,222.3 (b) Eigenvector for 21 you entered above: Vi = <-1/2,1> (c) Eigenvector for 22 you entered above: Part 2: Diagonalizability (d) Find a diagonal matrix D and an invertible matrix P D,...
4. (15 pts Consider the following direction fields IV VI (5 pts)Which of the direction fields corresponds to the system x -Ax, where A is a 2x2 matrix with eigenvalues λ,--1 and λ2-2 and corresponding eigenvectors vand v- 1? a. is a 2x2 matrix with repeated eigenvalue λ = 0 with defect 1 (has only one linearly independent eigenvector, not two.) and corresponding eigenvector vi- 13 (5 pts) Which of the direction fields corresponds to the system x -Cx, where...
Diagonalize the following matrix. The real eigenvalues are given to the right of the matrix. 1 -4 4 12 - 15 12 ; 2 = -3,5 16 - 16 13 Select the correct choice below and, if necessary, fill in the answer box to complete your choice. -3001 O A. For P= ,D= 050 | 005) -3 00 OB. For P= ,D= 0 -30 10 05 OC. The matrix cannot be diagonalized.
Diagonalize the following matrix. The real eigenvalues are given to the right of the matrix. 2 2 -4 - 1 5 -4 ; 2 = 3,8 -2 7 Select the correct choice below and, if necessary, fill in the answer box to complete your choice. O A. 3 0 0 For P = D= 0 3 0 0 0 8 (Simplify your answer.) B. 3 00 For P = D = 0 8 0 0 0 8 (Simplify your answer.)...
5. Consider the matrix A-O , where a, b, c are real numbers. In this problem, we consider what the са со dimensions of subspaces associated to A could be; these answers may depend on the values of a, b, and c, and we want to see what the possible dimensions are. (a) Could the null space of A be 0-dimensional? 1-dimensional 2-dimensional? (b) Could the column space of A be 0-dimensional? 1-dimensional? 2-dimensional? 10 marks] 5. Consider the matrix...
Diagonalize the following matrix. The real eigenvalues are given to the right of the matrix. 5 1 1 1 5 1 :1 = 4,7 1 1 5 Select the correct choice below and, if necessary, fill in the answer box to complete your choice. 400 D=070 O A. For Pa 0 0 7 4 0 0 OB. For Pa .D= 0 4 0 0 0 7 OC. The matrix cannot be diagonalized.
2. A 5 x 5 matrix of real numbers, A s found to have the following eigenvalues (a) Explain why A is, or is NOT, diagonalizable. (b Explain why A is, or is NOT, inzertibie. 2. A 5 x 5 matrix of real numbers, A s found to have the following eigenvalues (a) Explain why A is, or is NOT, diagonalizable. (b Explain why A is, or is NOT, inzertibie.