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MATH 557 Homework Set #6 Fall 2015 24. Find etA for the two cases 1 A = −4 −1 1 A= 0 0 −4 −1 −2 4 4 1 0 0 2 −1 0 2 25. Verify the statement made in class that if A is an n × n matrix for which |ajk | ≤ M for every j, k then |A| ≤ nM (here |A| is the matrix 2-norm). Show also that no better estimate is possible, i.e. there exists a matrix A with the stated properties, and a nonzero vector x for which |Ax| = nM |x|. 26. By transforming to an equivalent system and using the variation of parameters formula for systems, find a formula for the general solution of y 00 + y = g(t). (It should involve some kind of integral containing g.) Find the general solution explicitly when g(t) = tan t. 27. Let A(t) be a continuous matrix function on [0, ∞) and suppose that there exists a constant C1 > −∞ such that Z t tr(A(s)) ds ≥ C1 ∀t ≥ 0 0 Suppose also that Φ(t) is a fundamental matrix for x0 = A(t)x satisfying |Φ(t)| ≤ C2 for some constant C2 and all t ≥ 0. a) Show that there exists another constant C3 such that |Φ(t)−1 | ≤ C3 for all t ≥ 0. (Suggestion: think of Cramer’s rule.) b) Prove that no solution of x0 = A(t)x can satisfy |x(t)| → 0 as t → ∞ except x(t) ≡ 0.