All real values $a$ for a $2$-dimensional vector? Find all real numbers $a$ for which there exists a $2D$, nonzero vector $v$ such that:
$\begin{pmatrix} 2 & 12 \\ 2 & -3 \end{pmatrix} {v} = a {v}$.
I substituted $v$ with $\begin{pmatrix} c \\ d \end{pmatrix}$ and multiplied to obtain the system of equations: 
$2x+12y = ax$
$2x-3y= ay$
Since the value $a$ is only of importance, I added the two equations to obtain $4x+9y = ax + ay$. Would that mean that $a = 4, 9$ is correct? 

you can go from  $$2x+12y = kx,\, 2x-3y = ky $$ to $$\frac{y}{x} = \frac{k-2}{12} = \frac2{k+3}.$$  therefore $k$ satisfies the characteristic equation $$0=(k+3)(k-2) - 24 = k^2+k-30 = (k+6)(k-5).$$ therefore $$k = 5, -6 $$
