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Fall 2005 - Nagy's Class - Exam 2

Fall 2005 - Nagy's Class - Exam 2 - Print Name Student...

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Print Name: Student Number: Section Time: Math 20F. Midterm Exam 2 November 21, 2005 Read each question carefully, and answer each question completely. Show all of your work. No credit will be given for unsupported answers. Write your solutions clearly and legibly. No credit will be given for illegible solutions. 1. (6 points) Consider the matrices A = 1 2 3 - 1 0 1 , B = 2 3 1 1 . For each of the following expressions, compute it or explain why it is not defined. (a) A + A T , and B + B T . (b) AB and BA . (c) Find a 2 × 2 matrix C such that BC = 1 3 2 4 . (a) A + A T is not defined, because A is 2 × 3 and A T is 3 × 2. B + B T = 2 3 1 1 + 2 1 3 1 = 4 4 4 2 . (b) AB is not defined, because A is 2 × 3 and B is 2 × 2. BA = 2 3 1 1 1 2 3 - 1 0 1 = - 1 4 9 0 2 4 . (c) det( B ) = 2 - 3 = - 1, then B - 1 = ( - 1) 1 - 3 - 1 2 = - 1 3 1 - 2 . Then, C = B - 1 1 3 2 4 = - 1 3 1 - 2 1 3 2 4 = - 1 + 6 - 3 + 12 1 - 4 3 - 8 , that is, C = 5 9 - 3 - 5 .
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2. (6 points) Find the dimension and a basis for both the null space of A and the column space of A , where A = 1 - 3 - 8 - 3 - 2 4 6 0 0 1 5 7 .
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