exam.pdf - MAT342 Linear Algebra Midterm Exam#1 2/20-2014 No aids(notes books calculators etc are permitted In problems that are not multiple choice

exam.pdf - MAT342 Linear Algebra Midterm Exam#1 2/20-2014...

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MAT342 Linear Algebra Midterm Exam #1, 2/20-2014 No aids (notes, books, calculators, etc.) are permitted. In problems that are not multiple choice, every step in your arguments must be explained. In these problems, a correct answer without any reasoning gives 0 points. In the multiple choice exercises, on the other hand, you would not have to show your work. You get 1 pt by leaving a multiple choice problem blank (and 5 pts for a correct answer). 1. (40 pts) (a) What is the determinant of the following matrix? 0 2 - 1 5 0 0 3 0 2 - 3 9 2 0 1 - 3 2 A. -18 B. -6 C. -3 D. 3 E. 6 Solution: Score: 71% (b) Let A be a 3 × 3 -matrix with column vectors a 1 , a 2 , and a 3 . Let b = 5 a 1 + 2 a 3 = a 2 - 3 a 3 . Which one of the following statements is true? A. The system A x = b has no solution. B. The system A x = b has exactly one solution. C. The system A x = b has at most two solutions. D. The system A x = b has infinitely many solutions. E. We need more information to say how many solutions A x = b has. Solution: By reading off the coefficients in the linear combination, we find two solutions (5 , 0 , 2) and (0 , 1 , - 3) , and when there are at least two solutions, there must be infinitely many. Score: 92% (c) Which one of the following does not hold in general for invertible matrices A and B ? A. A + B = B + A B. AB = BA C. ( AB ) T = B T A T D. ( AB ) - 1 = B - 1 A - 1 E. ( A T ) - 1 = ( A - 1 ) T Solution: Score: 70% (20% picked A.) (d) Let a 1 , a 2 , . . . , a n and b be vectors in R m . What is the correct definition of b being a linear combination of a 1 , a 2 , . . . , a n ? A. b = a 1 + a 2 + · · · + a n . B. For some i there is a scalar c such that b = c a i . C. There exist scalars c 1 , c 2 , . . . , c n such that b = c 1 a 1 + c 2 a 2 + · · · + c n a n . D. b is the matrix having a 1 , a 2 , . . . , a n as its column vectors. This study resource was shared via CourseHero.com
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E. There exist scalars c 1 , c 2 , . . . , c n such that b is the matrix having c 1 a 1 , c 2 a 2 , . . . , c n a n as its column vectors.
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