Easy# Properties of the Eigenvalue Method

DIFFEQ-KVAGXK

Which of the following is always true about a system of linear differential equations $X'=AX$ where $A$ is an $n\times n$ matrix?

I) If $\lambda$ is an eigenvalue and $v$ is a corresponding eigenvector, then $X=ve^{\lambda t}$ is a solution to the system.

II) If $X_{1},X_{2},\cdots, X_{n}$ are linearly independent solutions to the system, then the general solution must be in the form $X=c_{1}X_{1}+c_{2}X_{2}+\cdots+c_{n}X_{n}$ for any constants $c_{1},c_{2},\cdots,c_{n}$.

III) If $X_{1},X_{2},\cdots,X_{n}$ are solutions to the system, then the Wroskian of $X_{1},X_{2},\cdots,X_{n}$ is not zero.