2u 712 I 22 7. (a) Let X 2(0),Y E M(C) be given. Suppose that for all a. 8 C 221 not both zero, rank (aX+BY) = 2. Show that y33 = 0. Give a 3 x 3 example that need not be of the form YO! where EM (C). 211 212 (b) Let X = 20. E M.(C) be given. Suppose that for all 0.8 C L21 not both zero, rank (a X + BY) = 2. Show that y = 0, 1, j = 3,...,n. (c) Deduce from (b) that: Suppose X = (21,22,0,...,01 Mn(C) and Y= ly1:42 ... , Yn! E M(C) such that rank(aX+BY) 2 for all a.3 EC not both zero. Then yt Espan{.21, 22}, j = 3,...,n. 20.30 T22 - 2u 712 I 22 7. (a) Let X 2(0),Y E M(C) be given. Suppose that for all a. 8 C 221 not both zero, rank (aX+BY) = 2. Show that y33 = 0. Give a 3 x 3 example that need not be of the form YO! where EM (C). 211 212 (b) Let X = 20. E M.(C) be given. Suppose that for all 0.8 C L21 not both zero, rank (a X + BY) = 2. Show that y = 0, 1, j = 3,...,n. (c) Deduce from (b) that: Suppose X = (21,22,0,...,01 Mn(C) and Y= ly1:42 ... , Yn! E M(C) such that rank(aX+BY) 2 for all a.3 EC not both zero. Then yt Espan{.21, 22}, j = 3,...,n. 20.30 T22