(3) In this question we introduce a new denition. Denition: We say that an n x 71 matrix A is slimy if: o A is diagonalizable; o A3 : A. 1 0 0 (a) Show that A : 0 1 0 is slimy. 0 0 1 (b) Because a slimy matrix is diagonalizablej we can write A : PDP'I, where D is diagonal and P is invertible. Prove that D3 : D, and so D is also slimy. (c) Prove that the only possible eigenvalues for a slimy matrix are 0, :I:1. ((1) Find all diagonal 2 X 2 slimy matrices. (e) Construct a nondiagonal 2 X 2 slimy matrix as follows: Choose any two nonparallel vectors v1,V2 E R2, but don't choose v1, v2 parallel to either e1,e2. Let P be the 2 x 2 matrix with columns v1 v2. Choose D - a diagonal slimy matrix from the previous part, but don't choose D = 12, -12, O2.. Compute PDP- and let A = PDP-1. (f) Explain why any matrix A constructed by the procedure in part (e) is slimy