# exponents and complex numbers

• Jul 6th 2010, 11:41 AM
jayshizwiz
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.
• Jul 6th 2010, 12:33 PM
Opalg
Don't be put off by the power 23. Start by thinking about the matrix $\displaystyle A = \begin{bmatrix}a&a^2+3\\-1&2\end{bmatrix}$. If A is invertible then so is $\displaystyle A^{23}$, which means that $\displaystyle A^{23}$ cannot send the nonzero vector $\displaystyle \begin{bmatrix}c\\1\end{bmatrix}$ to the zero vector. Therefore A is not invertible, which means that its determinant must be 0. If you write down the determinant of A and put it equal to 0 then you get a quadratic in a, which has two complex solutions. I guess those two values are what the question is asking for.
• Jul 6th 2010, 07:57 PM
jayshizwiz
Thanks! Great tips.