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MAT 275 Homework 6 Please submit your work of Homework 6 on Blackboard by Thursday, December 3rd by 11:59 pm. (1) Find the eigenvalues of   2 0 1 0 1 0 1 0 2  . (a) 1, 2. (b) 1, 3. (c) 0, −3, 1. (d) 0, 2, 1. (e) None of these. (2) Find the eigenvalues of   −2 1 1 1 −2 1 1 1 −2   . (a) 1, 2. (b) 0, 3. (c) 0, −3. (d) 0, 1. (e) None of these. 1

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MAT 275 Homework 6 Please submit your work of Homework 6 on Blackboard by Thursday, December 3rd by 11:59 pm. (1) Find the eigenvalues of   2 0 1 0 1 0 1 0 2  . (a) 1, 2. (b) 1, 3. (c) 0, −3, 1. (d) 0, 2, 1. (e) None of these. (2) Find the eigenvalues of   −2 1 1 1 −2 1 1 1 −2   . (a) 1, 2. (b) 0, 3. (c) 0, −3. (d) 0, 1. (e) None of these. 1 (3) Find the eigenvalues and corresponding eigenvectors of 3 −1 1 1 . (a) Only eigenvalue is 1 and corresponding eigenvector is nonzero scalar multiple of 1 0 . (b) Only eigenvalue is 0 and corresponding eigenvector is nonzero scalar multiple of 0 1 . (c) Only eigenvalue is 2 and corresponding eigenvector is nonzero scalar multiple of 1 1 . (d) Eigenvalues are 1, 0 and corresponding eigenvectors are nonzero scalar multiple of 1 0 , 0 1 respectively. (e) None of these. (4) Solve the initial value problem for y x 0 =x + 2y y 0 =2x + y x(0) =1, y(0) = −3. (a) y = −2e 2t − e 3t . (b) y = −2e t − e 3t . (c) y = −2e −t − e 3t . (d) y = −2e 2t − e t . (e) None of these. 2 (5) Solve the initial value problem for x x 0 =3x + y y 0 = − 2x x(0) =1, y(0) = 1. (a) x = −2e t + 3e −t . (b) x = e t − 3e 3t . (c) x = e t − 3e 2t . (d) x = −2e t + 3e 2t . (e) None of these. (6) Solve the initial value problem for y x 0 =x + 3y y 0 =2x + 2y x(0) =1, y(0) = −1 (a) y = 1 5 e 4t + 4 5 e −t . (b) y = −1 5 e 4t − 4 5 e −t . (c) y = −1 2 e t − e 2t . (d) y = e 4t + 2 3 e t . (e) None of these. 3 (7) Solve the initial value problem for x x 0 = − x + 3y y 0 = − 2x + 4y x(0) =1, y(0) = −1. (a) x = e t + 3e −t . (b) x = e 3t − 2e −t . (c) x = −1 2 e −t − e 2t . (d) x = e 2t . (e) None of these. (8) Solve the initial value problem for y x 0 =x + 2y y 0 =2x + y x(0) =2, y(0) = −1. (a) y = e 4t + 3e −t . (b) y = e 3t − 2e −t . (c) y = −3 2 e −t + 1 2 e 3t . (d) y = e −t + 2e 3t . (e) None of these. 4 (9) Solve the initial value problem for y x 0 =6x + 2y y 0 =2x + 3y x(0) =1, y(0) = 1. (a) y = 1 5 e 4t + 3e −7t . (b) y = 2 5 e 2t + 3 5 e 7t . (c) y = −3 4 e −t + 1 3 e 3t . (d) y = e −2t + 2e 7t . (e) None of these. (10) Can an eigenvector corresponding to the eigenvalue 0 be 0 0 ? (a) Yes. (b) No. (c) Not enough information. 5

 
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