Problem Set 33: Solving Systems with Inverses

1. In a previous section, we showed that matrix multiplication is not commutative, that is, [latex]AB\ne BA[/latex] in most cases. Can you explain why matrix multiplication is commutative for matrix inverses, that is, [latex]{A}^{-1}A=A{A}^{-1}?[/latex]

2. Does every [latex]2\times 2[/latex] matrix have an inverse? Explain why or why not. Explain what condition is necessary for an inverse to exist.

3. Can you explain whether a [latex]2\times 2[/latex] matrix with an entire row of zeros can have an inverse?

4. Can a matrix with an entire column of zeros have an inverse? Explain why or why not.

5. Can a matrix with zeros on the diagonal have an inverse? If so, find an example. If not, prove why not. For simplicity, assume a [latex]2\times 2[/latex] matrix.

In the following exercises, show that matrix [latex]A[/latex] is the inverse of matrix [latex]B[/latex].

6. [latex]A=\left[\begin{array}{cc}1& 0\\ -1& 1\end{array}\right],B=\left[\begin{array}{cc}1& 0\\ 1& 1\end{array}\right][/latex]

7. [latex]A=\left[\begin{array}{cc}1& 2\\ 3& 4\end{array}\right],B=\left[\begin{array}{cc}-2& 1\\ \frac{3}{2}& -\frac{1}{2}\end{array}\right][/latex]

8. [latex]A=\left[\begin{array}{cc}4& 5\\ 7& 0\end{array}\right],B=\left[\begin{array}{cc}0& \frac{1}{7}\\ \frac{1}{5}& -\frac{4}{35}\end{array}\right][/latex]

9. [latex]A=\left[\begin{array}{cc}-2& \frac{1}{2}\\ 3& -1\end{array}\right],B=\left[\begin{array}{cc}-2& -1\\ -6& -4\end{array}\right][/latex]

10. [latex]A=\left[\begin{array}{ccc}1& 0& 1\\ 0& 1& -1\\ 0& 1& 1\end{array}\right],B=\frac{1}{2}\left[\begin{array}{ccc}2& 1& -1\\ 0& 1& 1\\ 0& -1& 1\end{array}\right][/latex]

11. [latex]A=\left[\begin{array}{ccc}1& 2& 3\\ 4& 0& 2\\ 1& 6& 9\end{array}\right],B=\frac{1}{4}\left[\begin{array}{ccc}6& 0& -2\\ 17& -3& -5\\ -12& 2& 4\end{array}\right][/latex]

12. [latex]A=\left[\begin{array}{ccc}3& 8& 2\\ 1& 1& 1\\ 5& 6& 12\end{array}\right],B=\frac{1}{36}\left[\begin{array}{ccc}-6& 84& -6\\ 7& -26& 1\\ -1& -22& 5\end{array}\right][/latex]

For the following exercises, find the multiplicative inverse of each matrix, if it exists.

13. [latex]\left[\begin{array}{cc}3& -2\\ 1& 9\end{array}\right][/latex]

14. [latex]\left[\begin{array}{cc}-2& 2\\ 3& 1\end{array}\right][/latex]

15. [latex]\left[\begin{array}{cc}-3& 7\\ 9& 2\end{array}\right][/latex]

16. [latex]\left[\begin{array}{cc}-4& -3\\ -5& 8\end{array}\right][/latex]

17. [latex]\left[\begin{array}{cc}1& 1\\ 2& 2\end{array}\right][/latex]

18. [latex]\left[\begin{array}{cc}0& 1\\ 1& 0\end{array}\right][/latex]

19. [latex]\left[\begin{array}{cc}0.5& 1.5\\ 1& -0.5\end{array}\right][/latex]

20. [latex]\left[\begin{array}{ccc}1& 0& 6\\ -2& 1& 7\\ 3& 0& 2\end{array}\right][/latex]

21. [latex]\left[\begin{array}{ccc}0& 1& -3\\ 4& 1& 0\\ 1& 0& 5\end{array}\right][/latex]

22. [latex]\left[\begin{array}{ccc}1& 2& -1\\ -3& 4& 1\\ -2& -4& -5\end{array}\right][/latex]

23. [latex]\left[\begin{array}{ccc}1& 9& -3\\ 2& 5& 6\\ 4& -2& 7\end{array}\right][/latex]

24. [latex]\left[\begin{array}{ccc}1& -2& 3\\ -4& 8& -12\\ 1& 4& 2\end{array}\right][/latex]

25. [latex]\left[\begin{array}{ccc}\frac{1}{2}& \frac{1}{2}& \frac{1}{2}\\ \frac{1}{3}& \frac{1}{4}& \frac{1}{5}\\ \frac{1}{6}& \frac{1}{7}& \frac{1}{8}\end{array}\right][/latex]

26. [latex]\left[\begin{array}{ccc}1& 2& 3\\ 4& 5& 6\\ 7& 8& 9\end{array}\right][/latex]

For the following exercises, solve the system using the inverse of a [latex]2\times 2[/latex] matrix.

27. [latex]\begin{array}{l}\text{ }5x - 6y=-61\hfill \\ 4x+3y=-2\hfill \end{array}[/latex]

28. [latex]\begin{array}{l}8x+4y=-100\\ 3x - 4y=1\end{array}[/latex]

29. [latex]\begin{array}{l}3x - 2y=6\hfill \\ -x+5y=-2\hfill \end{array}[/latex]

30. [latex]\begin{array}{l}5x - 4y=-5\hfill \\ 4x+y=2.3\hfill \end{array}[/latex]

31. [latex]\begin{array}{l}-3x - 4y=9\hfill \\ 12x+4y=-6\hfill \end{array}[/latex]

32. [latex]\begin{array}{l}-2x+3y=\frac{3}{10}\hfill \\ -x+5y=\frac{1}{2}\hfill \end{array}[/latex]

33. [latex]\begin{array}{l}\frac{8}{5}x-\frac{4}{5}y=\frac{2}{5}\hfill \\ -\frac{8}{5}x+\frac{1}{5}y=\frac{7}{10}\hfill \end{array}[/latex]

34. [latex]\begin{array}{l}\frac{1}{2}x+\frac{1}{5}y=-\frac{1}{4}\\ \frac{1}{2}x-\frac{3}{5}y=-\frac{9}{4}\end{array}[/latex]

For the following exercises, solve a system using the inverse of a [latex]3\text{}\times \text{}3[/latex]
matrix.

35. [latex]\begin{array}{l}3x - 2y+5z=21\hfill \\ 5x+4y=37\hfill \\ x - 2y - 5z=5\hfill \end{array}[/latex]

36. [latex]\begin{array}{l}\text{ }4x+4y+4z=40\hfill \\ \text{ }2x - 3y+4z=-12\hfill \\ \text{ }-x+3y+4z=9\hfill \end{array}[/latex]

37. [latex]\begin{array}{l}\text{ }6x - 5y-z=31\hfill \\ \text{ }-x+2y+z=-6\hfill \\ \text{ }3x+3y+2z=13\hfill \end{array}[/latex]

38. [latex]\begin{array}{l}6x - 5y+2z=-4\hfill \\ 2x+5y-z=12\hfill \\ 2x+5y+z=12\hfill \end{array}[/latex]

39. [latex]\begin{array}{l}4x - 2y+3z=-12\hfill \\ 2x+2y - 9z=33\hfill \\ 6y - 4z=1\hfill \end{array}[/latex]

40. [latex]\begin{array}{l}\frac{1}{10}x-\frac{1}{5}y+4z=\frac{-41}{2}\\ \frac{1}{5}x - 20y+\frac{2}{5}z=-101\\ \frac{3}{10}x+4y-\frac{3}{10}z=23\end{array}[/latex]

41. [latex]\begin{array}{l}\frac{1}{2}x-\frac{1}{5}y+\frac{1}{5}z=\frac{31}{100}\hfill \\ -\frac{3}{4}x-\frac{1}{4}y+\frac{1}{2}z=\frac{7}{40}\hfill \\ -\frac{4}{5}x-\frac{1}{2}y+\frac{3}{2}z=\frac{1}{4}\hfill \end{array}[/latex]

42. [latex]\begin{array}{l}0.1x+0.2y+0.3z=-1.4\hfill \\ 0.1x - 0.2y+0.3z=0.6\hfill \\ 0.4y+0.9z=-2\hfill \end{array}[/latex]

For the following exercises, use a calculator to solve the system of equations with matrix inverses.

43. [latex]\begin{array}{l}2x-y=-3\hfill \\ -x+2y=2.3\hfill \end{array}[/latex]

44. [latex]\begin{array}{l}-\frac{1}{2}x-\frac{3}{2}y=-\frac{43}{20}\hfill \\ \frac{5}{2}x+\frac{11}{5}y=\frac{31}{4}\hfill \end{array}[/latex]

45. [latex]\begin{array}{l}12.3x - 2y - 2.5z=2\hfill \\ 36.9x+7y - 7.5z=-7\hfill \\ 8y - 5z=-10\hfill \end{array}[/latex]

46. [latex]\begin{array}{l}0.5x - 3y+6z=-0.8\hfill \\ 0.7x - 2y=-0.06\hfill \\ 0.5x+4y+5z=0\hfill \end{array}[/latex]

For the following exercises, find the inverse of the given matrix.

47. [latex]\left[\begin{array}{cccc}1& 0& 1& 0\\ 0& 1& 0& 1\\ 0& 1& 1& 0\\ 0& 0& 1& 1\end{array}\right][/latex]

48. [latex]\left[\begin{array}{rrrr}\hfill -1& \hfill 0& \hfill 2& \hfill 5\\ \hfill 0& \hfill 0& \hfill 0& \hfill 2\\ \hfill 0& \hfill 2& \hfill -1& \hfill 0\\ \hfill 1& \hfill -3& \hfill 0& \hfill 1\end{array}\right][/latex]

49. [latex]\left[\begin{array}{rrrr}\hfill 1& \hfill -2& \hfill 3& \hfill 0\\ \hfill 0& \hfill 1& \hfill 0& \hfill 2\\ \hfill 1& \hfill 4& \hfill -2& \hfill 3\\ \hfill -5& \hfill 0& \hfill 1& \hfill 1\end{array}\right][/latex]

50. [latex]\left[\begin{array}{rrrrr}\hfill 1& \hfill 2& \hfill 0& \hfill 2& \hfill 3\\ \hfill 0& \hfill 2& \hfill 1& \hfill 0& \hfill 0\\ \hfill 0& \hfill 0& \hfill 3& \hfill 0& \hfill 1\\ \hfill 0& \hfill 2& \hfill 0& \hfill 0& \hfill 1\\ \hfill 0& \hfill 0& \hfill 1& \hfill 2& \hfill 0\end{array}\right][/latex]

51. [latex]\left[\begin{array}{rrrrrr}\hfill 1& \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill 0\\ \hfill 0& \hfill 1& \hfill 0& \hfill 0& \hfill 0& \hfill 0\\ \hfill 0& \hfill 0& \hfill 1& \hfill 0& \hfill 0& \hfill 0\\ \hfill 0& \hfill 0& \hfill 0& \hfill 1& \hfill 0& \hfill 0\\ \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill 1& \hfill 0\\ \hfill 1& \hfill 1& \hfill 1& \hfill 1& \hfill 1& \hfill 1\end{array}\right][/latex]

For the following exercises, write a system of equations that represents the situation. Then, solve the system using the inverse of a matrix.

52. 2,400 tickets were sold for a basketball game. If the prices for floor 1 and floor 2 were different, and the total amount of money brought in is $64,000, how much was the price of each ticket?

53. In the previous exercise, if you were told there were 400 more tickets sold for floor 2 than floor 1, how much was the price of each ticket?

54. A food drive collected two different types of canned goods, green beans and kidney beans. The total number of collected cans was 350 and the total weight of all donated food was 348 lb, 12 oz. If the green bean cans weigh 2 oz less than the kidney bean cans, how many of each can was donated?

55. Students were asked to bring their favorite fruit to class. 95% of the fruits consisted of banana, apple, and oranges. If oranges were twice as popular as bananas, and apples were 5% less popular than bananas, what are the percentages of each individual fruit?

56. A sorority held a bake sale to raise money and sold brownies and chocolate chip cookies. They priced the brownies at $1 and the chocolate chip cookies at $0.75. They raised $700 and sold 850 items. How many brownies and how many cookies were sold?

57. A clothing store needs to order new inventory. It has three different types of hats for sale: straw hats, beanies, and cowboy hats. The straw hat is priced at $13.99, the beanie at $7.99, and the cowboy hat at $14.49. If 100 hats were sold this past quarter, $1,119 was taken in by sales, and the amount of beanies sold was 10 more than cowboy hats, how many of each should the clothing store order to replace those already sold?

58. Anna, Ashley, and Andrea weigh a combined 370 lb. If Andrea weighs 20 lb more than Ashley, and Anna weighs 1.5 times as much as Ashley, how much does each girl weigh?

59. Three roommates shared a package of 12 ice cream bars, but no one remembers who ate how many. If Tom ate twice as many ice cream bars as Joe, and Albert ate three less than Tom, how many ice cream bars did each roommate eat?

60. A farmer constructed a chicken coop out of chicken wire, wood, and plywood. The chicken wire cost $2 per square foot, the wood $10 per square foot, and the plywood $5 per square foot. The farmer spent a total of $51, and the total amount of materials used was [latex]14{\text{ ft}}^{2}[/latex]. He used [latex]{\text{3 ft}}^{2}[/latex] more chicken wire than plywood. How much of each material in did the farmer use?

61. Jay has lemon, orange, and pomegranate trees in his backyard. An orange weighs 8 oz, a lemon 5 oz, and a pomegranate 11 oz. Jay picked 142 pieces of fruit weighing a total of 70 lb, 10 oz. He picked 15.5 times more oranges than pomegranates. How many of each fruit did Jay pick?