I'm doing differential equations (systems of ODE)
The equation: $$\begin{bmatrix}2 & 7 \\-5 & -10\end{bmatrix}$$
I have found $\lambda$ to be -3 and -5
For $\lambda$ = -5 substituted into the 2x2 determinant matrix, we get
$$\begin{bmatrix}7 & 7 \\-5 & -5\end{bmatrix}$$
Now, I said $7x_1 = -7x_2 \implies x_1 = -x_2$ then I have the corresponding eigenvector of $$\begin{bmatrix}-1 \\1\end{bmatrix}$$ but the given solution has $$\begin{bmatrix}1 \\-1\end{bmatrix}$$
Therefore, part of my general solution for the equation is .... + $C_2e^{-5t}\begin{bmatrix}-1 \\1\end{bmatrix}$. Is that acceptable? Or is this wrong and the solution is right?
Thanks! TGIF