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Exam2_Math121 - Midterm 2 for Math 121 Fall 2006 Monday...

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Midterm 2 for Math 121, Fall 2006. Monday October 20. Time allowed: 53 minutes. You may assume all vector spaces are finite-dimensional unless otherwise stated. There are 110 points on this exam, and full score is 100 points. 1. Let A M 3 ( C ) be the 3 × 3 matrix 1 0 3 2 4 2 3 2 1 . (a) (40 points) Find the determinant, rank, characteristic polynomial, the eigen- values and the eigenvectors of A . Find an invertible matrix Q such that D = Q - 1 AQ is diagonal, and compute D . (Hint: all the eigenvalues are integers, and all the eigenvectors can be chosen to have integer coordinates.) (b) (15 points) Solve the following system of linear differential equations. x 1 ( t ) = x 1 ( t ) + 3 x 3 ( t ) x 2 ( t ) = 2 x 1 ( t ) + 4 x 2 ( t ) + 2 x 3 ( t ) x 3 ( t ) = 3 x 1 ( t ) + 2 x 2 ( t ) + x 3 ( t ) (c) (15 points) Let B be the matrix 1 / 6 0 1 / 2 1 / 3 2 / 3 1 / 3 1 / 2 1 / 3 1 / 6 . Compute lim m →∞ B m . (d) (10 points) Students at the three schools Harvard, MIT and Princeton can choose every year which school they will attend next year (they also never graduate). Students at Harvard will remain at Harvard with 1/6 probability,
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