Matrix Algebra Question (Linear Algebra) Find all values of $a$ such that $A^3 = 2A$, where
$$A = \begin{bmatrix} -2 & 2 \\ -1 & a \end{bmatrix}.$$
The matrix I got for $A^3$ at the end didn't match up, but I probably made a multiplication mistake somewhere.

The matrix $A$ is clearly never a multiple of the identity, so its minimal polynomial is of degree$~2$, and equal to its characteristic polynomial, which is $P=X^2-(a-2)X+2-2a$. Now the polynomial $X^3-2X$ annihilates $A$ if and only if it is divisible by the minimal polynomial $P$. Euclidean division of $X^3-2X$ by $P$ gives quotient $X+a-2$ and remainder $(a^2-2a)X-(a-2)(2-2a)$. This remainder is clearly$~0$ if $a=2$ and for no other values of $a$, so $A^3=2A$ holds precisely for that value.
You can in fact avoid doing the Euclidean division by observing, since $X$ divides $X^3-2X$, that for $P$ to divide $X^3-2X$, unless $P$ itself has a factor $X$ (which happens for $a=1$ but divisibility then fails anyway), $P$ must divide (and in fact be equal to) $(X^3-2X)/X=X^2-2$; this happens for $a=2$.
