Chapter 9 - Systems of Differential Equations - 9.4 Vector Differential Equations: Nondefective Coefficient Matrix - True-False Review - Page 607: a

True

If $(\lambda, v)$ is an eigenvalue-eigenvector pair for $A$, then to prove that $x(t)=e^{\lambda t}v$ is a solution to the vector differential equation, we obtain: $\rightarrow x'(t)=e^{\lambda t}(\lambda v)=e^{\lambda t}(Av)=A(e^{\lambda t}v)=Ax(t)$ Hence, the statement is true.