Exercise 5.4.17

If we have the characteristic polynomial to be

then we have

by Cayley-Hamilton Theorem. This means that $A^n$ is a linear combination of $I,A,A^2,\dots ,A^{n-1}$. By multiplying both sides by $A$, we know that $A^{n+1}$ is a linear combination of $A,A^2,\dots ,A^n$. Since $A^n$ can be represented as a linear combination of previous terms, we know that $A^{n+1}$ could also be a linear combination of $I,A,A^2,\dots ,A^{n-1}$. Inductively, we know that

and so the dimension could not be greater than $n$.