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Exam 3 Review - MATH 3321 Sample Questions for Exam 3 2x 4...

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MATH 3321 Sample Questions for Exam 3 1. Find x and y so that 2 x 4 - 3 5 x + 3 y - 2 - 2 - y = - 5 2 - 5 12 . 2. Let A = 2 - 1 3 0 4 - 2 , B = - 3 1 2 5 , C = 3 - 2 0 - 1 1 2 . Perform the indicated operations, if possible: (a) AC (b) AB (c) B + AC (d) CBA 3. Show that the matrix of coefficients is non-singular and use Cramer’s rule to find the unique solution of the system: x 1 - x 2 + x 3 = 1 2 x 2 - x 3 = 1 2 x 1 + 3 x 2 = 1 4. Write the system in the vector-matrix form A x = b and solve by finding A - 1 . 2 x 1 + x 2 = 2 4 x 1 + 3 x 2 = - 4 5. Write the system in the vector-matrix form A x = b and solve by finding A - 1 . x 1 + x 2 = 2 2 x 1 + 3 x 2 - x 3 = 0 x 1 + 2 x 3 = 4 6. Given the set of vectors { u = ( 1 , - 2 , 1 ) , v = ( 2 , 1 , - 1 ) , w = ( 7 , - 4 , 1 ) } 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. 7. Given the set of vectors { v 1 = ( 2 , 0 , - 1 ) , v 2 = ( - 3 , 1 , 2 ) , v 3 = ( 8 , - 2 , - 5 ) , v 4 = ( - 9 , 1 , 5 ) } Is the set linearly dependent or linearly independent? If it is linearly dependent, how many independent vectors are there in the set? 1
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8. For what values of a are the vectors v 1 = ( a, 1 , - 1 ) , v 2 = ( - 1 , 2 a, 3 ) , v 3 = ( - 2 , a, 2 ) , v 4 = ( 3 a, - 2 , a ) linearly dependent?
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