Answer Happy

1. Suppose that A= -1 5 -10 14 . Do each of the following: i) ii) iii) iv) Find a diagonal form for A, that is, find invertible matrix P and diagonal matrix D such that A = PDP-. It is vital you do this part correctly or the rest of the problem is shot. Note that A” = P(D")p- for all natural numbers n . Illustrate this by proving the cases when n=2 and n=3. ? ? What is the matrix D'anyway? Fill in the entries: D" ? ? 1 Does the formula for D" in part iii above hold if, say, n== ? That is the question 2 you will answer by doing the following: As a natural definition, we want D12 to satisfy (012) = D. Can you find any matrices that will satisfy this last condition? You should find FOUR such matrices (Hint: since we are attempting to satisfy the equation X2 = D , consider both + entries in a matrix; we are not confining ourselves to non-negative square roots like we usually do when talking about a square root). 1 Does the formula in part ii above hold when n=-? Find the four "square roots” 2 of A by plugging in the "square roots” D12 you found from part iv above into the formula for A" from part ii with and compute. You might check to see if the matrices you came up with actually equal the matrix A when squared. v)