The trace of a square matrix is the sum of its diagonal entries.
(a) Find the formula for the characteristic polynomial of a 2 × 2 matrix.
(b) Show that trace is invariant under similarity, and so we can sensibly speak of the 'trace of a map'. (Hint: see the prior item.)
(c) Is trace invariant under matrix equivalence?
(d) Show that the trace of a map is the sum of its eigenvalues (counting multiplicities).
(e) Show that the trace of a nilpotent map is zero. Does the converse hold?