All solutions must be supported with a proof, a counterexample, or a detailed computation, as appropriate. Solutions should be written incomplete sentences and paragraphs.

Let a. b A - (a d) be a 2 x 2 matrix over IF, and let T : F2 F2 be the linear transformation Tv = Av. (a) Show that T is invertible if and only if ad be 75 0. (b) When T is invertible, compute NWT)"1 with respect to the standard bases. (c) Compute the eigenvalue(s) of T in terms of a,b,c,d. Discuss how many eigenvalues there are depending on cub, c,d. Your answer will depend on whether IF = R or III. (d) (Optional challenge) Now suppose IF = C. Show that exactly one of the following holds: c There is a basis of C2 and numbers Ahhg E C (possiny equal) such that the matrix of T with respect to this basis (on both the domain and codomain) is _ A1 0 Mm _ (0 A2) . 0 There is a basis of C2 and a number A E III such that M(T) = (3 '1')