Let A be a complex n × n matrix. Prove that if A is invertible and if the eigenvalues of A are (A1,..., An), then the eigenvalues of A-' are (A¬',.,

Let P denote a matrix whose columns are eigenvectors K,, K2, ..., K, corresponding to distinct eigenvalues 1,, 1,,..., , of an nxn matrix A. Then it can be shown that A = PDP-1, where D is a diagonal matrix defined by the following. D = (9) …… 入。 If D is defined as in (9), then find e Dt eDt

Show that if 0 is an eigenvector of A, then A is singular.