(a) Prove that a Jordan block matrix J0,n with zero diagonal entries is nilpotent. as in Exercise 1.3.13.
(b) Prove that a Jordan matrix is nilpotent if and only if all its diagonal entries are zero.
(c) Prove that a matrix is nilpotent if and only if its Jordan canonical form is nilpotent.
(d) Explain why a matrix is nilpotent if and only if its only eigenvalue is 0.