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MATH2270-004 Name: Midterm 3 Answers November 20, 2015 Instructions: Do all the problems on both sides of each page. Show all your work and box your answers. If you get stuck on a problem, skip it and come back to it at the end. h1 0 0i 1. (a) 0 2 0 003 (b) λ = 1, 2, 3 2. [ 40 14 ] 3. (a) Yes h1i (b) 1 7 h 2 i (c) −2 0 1 h1i 1 h 1 i 1 h 1 i 1 , √ −1 (d) √ 1 , √ 3 1 6 −2 2 0 1 4. ~x(t) = c1 e2t [ 11 ] + c2 e−2t −3 5. (a) xk = 1.1k [ 13 ] + .5k [ 21 ] (b) Origin is a saddle. Draw a picture to show trajectories. h −1 i 6. (a) 1 1 h0i (b) 1 0 7. (c) No. There is no basis of eigenvectors. 2h 6 i 7 −2 3 8. No. There is no basis of eigenvectors since the sum of the dimensions of the eigenspaces of A is less than 6. 9. [ 21 21 ] Any matrix with linearly dependent columns will do (as long as no entries are 0). 10. Expanding in the first row, we get (2 − λ) [(4 − λ)(2 − λ) + 1] + 1 [(2 − λ) + 1] − 1 [−1 + (4 − λ)] 1