Exam (elaborations) TEST BANK FOR Elementary Linear Algebra with Applications 9th Edition By Kolman, Hill (Solution manual)

Linear Equations and Matrices Section 1.1, p. 8 2. x = 1, y = 2, z = −2. 4. No solution. 6. x = 13 + 10t, y = −8 − 8t, t any real number. 8. Inconsistent; no solution. 10. x = 2, y = −1. 12. No solution. 14. x = −1, y = 2, z = −2. 16. (a) For example: s = 0, t = 0 is one answer. (b) For example: s = 3, t = 4 is one answer. (c) s = t2 . 18. Yes. The trivial solution is always a solution to a homogeneous system. 20. x = 1, y = 1, z = 4. 22. r = −3. 24. If x1 = s1, x2 = s2, . . . , xn = sn satisfy each equation of (2) in the original order, then those same numbers satisfy each equation of (2) when the equations are listed with one of the original ones interchanged, and conversely. 25. If x1 = s1, x2 = s2, . . . , xn = sn is a solution to (2), then the pth and qth equations are satisfied. That is, ap1s1 + · · · + apnsn = bp aq1s1 + · · · + aqnsn = bq. Thus, for any real number r, (ap1 + raq1)s1 + · · · + (apn + raqn)sn = bp + rbq. Then if the qth equation in (2) is replaced by the preceding equation, the values x1 = s1, x2 = s2, . . . , xn = sn are a solution to the new linear system since they satisfy each of the equations. 2 Chapter 1 26. (a) A unique point. (b) There are infinitely many points. (c) No points simultaneously lie in all three planes. 28. No points of intersection: C1 C2 C2 C1 One point of intersection: C1 C2 Two points of intersection: C1 C2 Infinitely many points of intersection: C1= C2 30. 20 tons of low-sulfur fuel, 20 tons of high-sulfur fuel. 32. 3.2 ounces of food A, 4.2 ounces of food B, and 2 ounces of food C. 34. (a) p(1) = a(1)2 + b(1) + c = a + b + c = −5 p(−1) = a(−1)2 + b(−1) + c = a − b + c = 1 p(2) = a(2)2 + b(2) + c = 4a + 2b + c = 7. (b) a = 5, b = −3, c = −7. Section 1.2, p. 19 2. (a) A = ! """""# 0 1 0 0 1 1 0 1 1 1 0 1 0 0 0 0 1 0 0 0 1 1 0 0 0 $ %%%%%& (b) A = ! """""# 0 1 1 1 1 1 0 1 0 0 1 1 0 1 0 1 0 1 0 0 1 0 0 0 0 $ %%%%%& . 4. a = 3, b = 1, c = 8, d = −2. 6. (a) C + E = E + C = ! # 5 −5 8 4 2 9 5 3 4 $ &. (b) Impossible. (c) ' 7 −7 0 1 ( . (d) ! # −9 3 −9 −12 −3 −15 −6 −3 −9 $ &. (e) ! # 0 10 −9 8 −1 −2 −5 −4 3 $ &. (f) Impossible. 8. (a) AT = ! # 1 2 2 1 3 4 $ &, (AT )T = ' 1 2 3 2 1 4 ( . (b) ! # 5 4 5 −5 2 3 8 9 4 $ &. (c) ' −6 10 11 17