The question is:
Suppose there is a $3$-dimension vector space $V$ over the field of real number $R$, and there is a $3\times3$ matrix $A$ whose entries are real numbers. $A$ is the matrix representation of a linear transformation $\mathscr{A}$ under some basis of $V$. What dimension of invariant subspaces must it has? Choices are $1, 2, 3$.
Here is what I think:
Since the characteristic polynomial of $A$ is of degree $3$, it must have a root in $R$, thus it has a characteristic subspace of dimension $1$, which is also its invariant subspace. And obviously $V$ is its invariant subspace, thus $3$ is right.
But I do not know how to check whether it must have a $2$-dimensional invariant subspace. How can I proceed from this? If possible, I'm also very willing to know how to deal with similar problem for $4\times4$ matrix or higher dimension. Thank you!