For each of the following, provide a proof or counterexample.

(1) If $A, B$ are complex $n \times n$ matrices and $A B=B A$, then $A$ and $B$ have a common eigenvector.

(2) If $A, B$ are complex $n \times n$ matrices and $A B=B A$, then $A$ and $B$ have a common eigenvalue.

(3) If $A, B$ are complex $n \times n$ matrices and $(A B)^{n}=0$ then $(B A)^{n}=0$.

(4) If $T: V \rightarrow V$ is an endomorphism of a finite-dimensional vector space $V$ and $\lambda$ is an eigenvalue of $T$, then the dimension of $\{v \in V \mid(T-\lambda I) v=0\}$ equals the multiplicity of $\lambda$ as a root of the minimal polynomial of $T$.

(5) If $T: V \rightarrow V$ is an endomorphism of a finite-dimensional complex vector space $V$, $\lambda$ is an eigenvalue of $T$, and $W_{i}=\left\{v \in V \mid(T-\lambda I)^{i}(v)=0\right\}$, then $W_{c}=W_{c+1}$ where $c$ is the multiplicity of $\lambda$ as a root of the minimal polynomial of $T$.