Answer
False
Work Step by Step
$A$ and $B$ are $n \times n$ matrices with the same characteristic equation
Let $A=\begin{bmatrix}
0 & 0\\
0 & 0
\end{bmatrix}$
$B=\begin{bmatrix}
0 & 1\\
0 & 0
\end{bmatrix}$
then for both $\lambda^2=0$
The constant vector function $x_1(t)= \begin{bmatrix}
c_1\\
c_2
\end{bmatrix}$ is a solution to $Ax=x'$
while $x_2(t)= \begin{bmatrix}
c_2\\
0
\end{bmatrix}=0\\
c_2=0$ is a solution to $Bx=x'$
Hence the solution sets to the vector differential equations $x′ = Ax$ and $x′ = Bx$ are not the same.
