Let be nilpotent matrices such that . Evaluate the determinant
Lemma: If and are nilpotent matrices that commute and are scalars, then is nilpotent.
Proof: Since and commute, they are simultaneously triangularizable. Let be an invertible matrix such that and , where and are upper triangular. Note that since and are nilpotent, and must have zeros down the main diagonal. Hence is upper triangular with zeros along the main diagonal which means that it’s nilpotent. Finally and so is nilpotent.
We have . Then equating the two left hand sides and simplifying gives us . Thus by the lemma we know that is nilpotent, i.e., it’s eigenvalues are all zero. It follows that the eigenvalues of are all one and so .