What would be the dimension of the vector space $W$?

Let $C$ be $n \times n$ matrix over the real number field $\mathbb{R}$.

Let $W$ be the vector space spanned by the matrices

$$I, C,C^2,C^3, \ldots,C^{(2n)}.$$

What is the dimension of the vector space $W$ ?

Following the hint below, $C$ satisfies its own characteristic equation $$a_0I+a_1C+\cdots+a_nC^n=0.$$ So $\{I,C,C^2, \cdots, C^n \}$ is linearly dependent set and hence the rest matrices $C^{n+1}, C^{n+2}, \cdots, C^{2n}$ generate a vector space of dimension at most $n$.

So the answer should be atmost $n$.

Solution 1:

HINT:

Every square matrix satisfies its own characteristic equation.