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.