In this problem, you'll work with some new definitions, one of which you've actually seen before. Before attempting the problem, read the definitions carefully and recall some of the work you've done with one of these definitions on a previous assignment and tutorial. It's important to get a feel for new definitions before you start answering problems about them. Definition 1 Let A be an nn matrix. - We define A to be happy if A2=A. - We define A to be melancholy if A2=0n. - We define A to be nihilistic if there is an integer k1 such that Ak=0n. (Recall that 0n is the nn matrix of all zeroes.) For each part, determine if you believe the statement is true or false, and justify your answers as appropriate. (a) Find a 22 melancholy matrix, not all of the entries of which are zero. (b) Find a matrix that is nihilistic but not melancholy. (c) Show that if an nn matrix A is happy, then all of its eigenvalues are either 0 or 1 . (d) True or false: If an nn matrix A is nihilistic, then 0 is an eigenvalue of A. (e) True or false: If an nn matrix A is happy, then both 0 and 1 are always eigenvalues of A