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MATH 3321 Sample Questions for Exam 3 1. Find x and y so that ± 2 x 4 - 35 x ² + ± 3 y - 2 - 2 - y ² = ± - 52 - 51 2 ² . Answer: x =31 / 17 ,y = - 49 / 17 2. Let A = ± 2 - 13 04 - 2 ² ,B = ± - 31 25 ² ,C = 3 - 2 0 - 1 12 . Perform the indicated operations, if possible: (a) AC (b) AB (c) B + AC (d) CBA Answer: AB is not de±ned, the others are de±ned. 3. Show that the matrix of coeﬃcients is non-singular and use Cramer’s rule to ±nd the unique solution of the system: x 1 - x 2 + x 3 =1 2 x 2 - x 3 2 x 1 +3 x 2 Answer: x =5 = - 3 ,z = - 7 4. Write the system in the vector-matrix form A x = b and solve by ±nding A - 1 . 2 x 1 + x 2 =2 4 x 1 x 2 = - 4 Answer: x 1 ,x 2 = - 8 ,A - 1 = ± 3 / 2 - 1 / 2 - 21 ² 5. Write the system in the vector-matrix form A x = b and solve by ±nding A - 1 . x 1 + x 2 2 x 1 x 2 - x 3 =0 x 1 +2 x 3 =4 Answer: x 1 =8 2 = - 6 3 = - 2 A - 1 = 6 - 2 - 1 - 521 - 311 1

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6. Given the set of vectors { u =(1 , - 2 , 1) , v =(2 , 1 , - , w =(7 , - 4 , } Is the set linearly dependent or linearly independent? If it is linearly dependent, express one of the vectors as a linear combination of the other two. Answer: Dependent, w =3 u +2 v 7. Given the set of vectors { v 1 , 0 , - , v 2 =( - 3 , 1 , 2) , v 3 =(8 , - 2 , - 5) , v 4 - 9 , 1 , } Is the set linearly dependent or linearly independent? If it is linearly dependent, how many independent vectors are there in the set? Answer: Dependent, 3 8. For what values of a are the vectors v 1 a, 1 , - , v 2 - 1 , 2 a, 3) , v 3 - 2 ,a , , v 4 =(3 a, - 2 ) linearly dependent? Answer: All real numbers. 9. For what values of a are the vectors u a, 1 , - , v - 1 , 2 a, , w - 2 , linearly dependent?
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