MATH 21B Midterm: MATH 21B Harvard 21b Fall 07first Midterm Practice2 Solution
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If needed, use the back or the next empty page for work. If you need additional paper, write your name on it: do not detach pages from this exam packet or un-staple the packet, please write neatly. Answers which are illegible for the grader can not be given credit: no notes, books, calculators, computers, or other electronic aids can be allowed, you have 90 minutes time to complete your work. Problem 1) tf questions (20 points) no justi cations needed. If a is invertible, then a3 is invertible. One could also see it by looking at the kernel: if a3~v = ~0 with a nonzero ~v, then a(a2(~v)) = a~u = ~0 and a had a nontrivial kernel and could not be invertible. If a2 is invertible, then a is invertible. If a2 is invertible then a2 has full rank, but then also a has full rank.
