Proving Linear Dependence in Complex Matrix Space

[Monochrome]

Homework Statement

If C is a n by n matrix with complex coefficient show that there is an integer k >0 that depends only on n such that C, C^2,C^3,...C^k are lin dependant.

The Attempt at a Solution

Its meant to be a full proof but the only idea I vaguely have is that i^4=i. I'm sure this is something standard to prove but I couldn't find it on google, so a link with an explanation would be appreciated.

 

[Benny]

I can't remember much linear algebra but I would try considering the following. Suppose the dimension of a vector space is 6. Is it possible to extract 7 linearly independent vectors from that vector space?

 

[morphism]

Cayley-Hamilton tells us that C satisfies its own characteristic polynomial.