I have the following system of differential equations:
$$\left\{\begin{aligned} \frac {dx} {dt}=-4x+2y \\ \frac {dy} {dt}=-\frac 5 2x+2y \end{aligned} \right. $$
Which corresponds to the following augmented matrix (to solve for eigenvectors)
$$ \begin{pmatrix} -4 & 2 \\ -\frac 5 2 & 2\\ \end{pmatrix} \begin{pmatrix} x \\ y\\ \end{pmatrix} = \begin{pmatrix} 0 \\ 0\\ \end{pmatrix} $$
The first thing I did was to get rid of the fraction by multiplying the second row by 2.
$$ \begin{pmatrix} -4 & 2 \\ -5 & 4\\ \end{pmatrix} \begin{pmatrix} x \\ y\\ \end{pmatrix} = \begin{pmatrix} 0 \\ 0\\ \end{pmatrix} $$
To find the eigenvectors, I start by getting the determinant of the following matrix :
$$ det\begin{pmatrix} -4-\lambda & 2 \\ -5 & 4-\lambda\\ \end{pmatrix} $$
Which leads to this: $\lambda^2 -6$. But this of course is not correct! The solution says that I'm suppose to end up with $\lambda^2 +2\lambda-3$, giving me eigenvalues $\lambda_1=1, \lambda_2=-3$. The only way to end up with a polynomial like that is to NOT multiply the second row by two. So how do I go about doing something like this? Am I missing something here?