Thread: exponents and complex numbers

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

Given - is a complex number that fulfills

is a complex number

What are all the possible values of ?

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.

Don't be put off by the power 23. Start by thinking about the matrix . If A is invertible then so is , which means that cannot send the nonzero vector 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.

Thanks! Great tips.