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Math Help - exponents and complex numbers

  1. #1
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    exponents and complex numbers

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

    Given - c is a complex number that fulfills \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)

    a is a complex number

    What are all the possible values of 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.
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  2. #2
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    Don't be put off by the power 23. Start by thinking about the matrix A = \begin{bmatrix}a&a^2+3\\-1&2\end{bmatrix}. If A is invertible then so is A^{23}, which means that A^{23} cannot send the nonzero vector \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.
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  3. #3
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    Thanks! Great tips.
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