Answer
False
Work Step by Step
If $A$ is an $n \times n$ defective matrix, then the vector differential equation $x'=Ax$ has $n$ linearly independent solutions regardless of any condition on $A$.
Differential Equations and Linear Algebra (4th Edition)
by Goode, Stephen W.; Annin, Scott A.
Published by
Pearson
ISBN 10:
0-32196-467-5
ISBN 13:
978-0-32196-467-0
Chapter 9 - Systems of Differential Equations - 9.5 Vector Differential Equations: Defective Coefficient Matrix - True-False Review - Page 618: a
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