Need help from 2-12

(2) If z, w E C, show that Re(zw) = Re(z) Re(w) - Im(z) Im(w). (Hint: write z = a + bi and w = c + di, then multiply out zw.) Note: it is also true that Im(zw) - Re(z) Im(w) + Im(z) Re(w), though you don't have to show that on this problem set. (3) Using the identities for Re(zw) and Im(zw) from problem 2, plug in z = e201 and w - e202. Simplify using elt = cos(x) + isin(x) - what do you get? Can you obtain the double angle trig identities for sine and cosine? (4) Find the reduced row echelon form of the matrix 4- 127 -2 : I (5) Find the determinant of the matrix 2+2 3-27 2 A = i 16 1 3 + 27] (6) (a) If z is a complex number, show that Re(z) = ? (z + z). (Hint: write z = a + bi.) (b) Use this to explain why Re() E Span(v, v). (Hint: part (a) suggests how to write Re(v) as a linear combination of v and v.) You can do something similar for Im(z) and Im(v), though you don't have to do that for this problem set. (7) Find the eigenvalues of the matrix A = (8) If A = , , find an invertible matrix P and a diagonal matrix D, possibly with complex entries, such that A = PDP-1. (9) If A= 2 , , find an invertible matrix P and a diagonal matrix D, possibly with complex entries, such that A = PDP-1 (10) If A is the matrix in problem 8, what is A5? (11) If A is a 2 x 2 matrix, 1 = a + bi is an eigenvalue of A, and b / 0, explain why A is diagonalizable. (12) Let A be a 3 x 3 matrix. (a) Explain why A has at least one eigenvalue in R. (b) If A has one eigenvalue in R and two eigenvalues not in R, explain why A is diagonalizable