The trace of a nilpotent matrix over a field is 0

Let N be an n \times n matrix over a field F, and suppose N^k = 0 for some k. Show that \mathsf{trace}\ N = 0.


We showed previously that the eigenvalues of N are all 0. We also showed previously that the trace of a matrix is the sum of its eigenvalues. Thus N has trace 0.

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