Proving controllability using eigenvectors and eigenvalues
Solution 1:
If more than one linearly independent eigenvector can be associated with a single eigenvalue $\lambda$ of $A$ then the rank of $\lambda\,I-A$ drops by more than one, since those eigenvectors span the null space of $\lambda\,I-A$. However, when using the PBH test and that $b$ has only one column it can be noted that $b$ can at most increase the rank of $[\lambda\,I-A, b]$ by one, which implies rank-deficiency and thus uncontrollability.