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Math 2270 Exam 1 - University of Utah Fall 2012 Name: (ey This is a 50 minute exam. Please show all your work, as a worked problem is required for full points, and partial credit may be rewarded for some work in the right direction. 1. (15 points) Vector Basics For the vectors arrr( 1) b=f i) 4J \1) /1 c=f 2 answer the following, or explain why the question does not make sense: (a) (3 points) 2a + 3c i( = (D (‘)f() 1 i) 4) arrf b=( ii /1 c=( 2 1J (b) (3 points) aH l+I6 (c) (2 points) What are the components of a unit vector in the same direction as a? / / 2 c) N cD. N 0 Q II + H r T 0 I. II I’ I 2. (10 points) Matrix Basics For the matrices (3 4 2 A 2 1 1 B= /2 1 5 4 4 2 1 02 ( 1 1 0 0 answer the following, or explain why the question does not make sense: (a) (3points)A+Crr (3 - ttj 16oO 4 cDCC CDCN II (z C r\rJ — C Cj c) 0 0 ‘-Th 1 c’- Lfl 3. (15 points) Elimination Issues (a) (5 points) For what value of a in the system of equations below does elimination fail to produce a unique solution? 3x + 2y 6x + ay = = ir 11 /c{ Ji1 ( a e C 10 b rc ie4 j 1 ru Z 1/ /ce; q h- /e p O Oy (b) (5 points) Given the determined value of a, for what value of b are there an infinite number of solutions? T b?o 7(j o Oy a(ty1 s (c) (5 points) For the determined values of a and b what are two dis tinct solutions? / 0 y Thte / qfN/ o4t 6 (QL/J 4. (20 points) Systems of Equations Use elementary row operations to convert the system of equations 2r 6r 12x + + + 3y + 6y + 9y — 3z = 12z z = 3 13 2 into upper-triangular form, and then use back-substitution to solve for the variables x, y, z. Be sure to show all your work. 5 (Rc 1Q-/ 5 L + t1 Z Y 3 fc(J3 Z\4Jy 3y LI y 3 - rc, /J (f or/ 7 x a) —c —QQ I oç — — — — — - c_J _ , H C a) — 0 ——H _Lf — a) -d co I ---- — 11 JZ cx-) c) >< ct 0) C 0 U J —r 0 UN Ui fl 7. (10 points) Symmetric Products For the matrix 2 3 4 0 (a) (4 points) What is the transpose RT? (b) (4 points) What is the symmetric product RTR 73 1 LC) 30/ (c) (2 points) Does RTR = RRT? / he 10