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MATH 152 REVIEW PROBLEMS (1) Suppose that T : R2 → R3 is a linear transformation satisfying 2 1 T = −1 4 3 0 2 T = 4 6 −2 Find T [ 01 ]. (2) Given the transition matrix 3 4 1 4 3 4 1 4 , find the equilibrium probability. (3) Find the determinant of the matrix 0 1 7 5 0 0 0 6 . 0 0 −3 3 2 −1 4 0 (4) Find the inverse of 1 2 3 0 1 2 . 0 0 1 (5) Let A be given by the matrix 25 0 0 0 35 0 . 0 0 30 (a) What is det A? (b) What are the eigenvalues of A? (c) What is A−1 ? (6) Find all vectors in R4 that are orthogonal to (1, 2, −2, 3), (−2, 3, 5, −1), and (−4, 13, 11, 3). 1 2 MATH 152 REVIEW PROBLEMS (7) If A has eigenvalues and eigenvectors given by λ1 = 1 k~1 = (1, 0, 5)T λ2 = 0 k~2 = (0, 1, 1)T λ3 = k~3 = (0, 1, −1)T 1 2 (a) Write (1, 0, 0)T as a linear combination of eigenvectors. (b) What is A5 (1, 0, 0)T ? ~ (c) What is the solution X(t) to the system of linear differential equations given by ~ dX ~ = AX dt ~ with X(0) = (1, 0, 0)T ? (8) Consider the system given by d~x = A~x dt (1) where A has eigenvalues and eigenvectors λ1 = 1 k~1 = (1, 0, 1)T λ2 = 2 + i k~2 = (i, 1, 0)T λ3 = 2 − i k~3 = (−i, 1, 0)T . (a) Write the general solution to (1). (b) Find the solutions of (1) (written in terms of real functions of t) that satisfy the initial conditions ~x(0) = (1, 2, 3)T .