exponents and complex numbers

Hi. I'm not sure where this question belongs so sorry if it doesn't belong here.

Given - $\displaystyle c$ is a complex number that fulfills $\displaystyle \left(\begin{array}{cc}a&{a^2+3}\\{-1}&2\end{array}\right)^{23} \left(\begin{array}{cc}c\\1\end{array}\right) = \left(\begin{array}{cc}0\\0\end{array}\right)$

$\displaystyle a$ is a complex number

What are all the possible values of $\displaystyle a$?

I'm not sure how to approach the question. What do I do if I have a matrix^23? And how do I know if the result times the next matrix will equal 0?

Thanks.