## Section Exercises

1. Explain why we can always evaluate the determinant of a square matrix.

2. Examining Cramer’s Rule, explain why there is no unique solution to the system when the determinant of your matrix is 0. For simplicity, use a $2\times 2$ matrix.

3. Explain what it means in terms of an inverse for a matrix to have a 0 determinant.

4. The determinant of $2\times 2$ matrix $A$ is 3. If you switch the rows and multiply the first row by 6 and the second row by 2, explain how to find the determinant and provide the answer.

For the following exercises, find the determinant.

5. $|\begin{array}{cc}1& 2\\ 3& 4\end{array}|$

6. $|\begin{array}{rr}\hfill -1& \hfill 2\\ \hfill 3& \hfill -4\end{array}|$

7. $|\begin{array}{rr}\hfill 2& \hfill -5\\ \hfill -1& \hfill 6\end{array}|$

8. $|\begin{array}{cc}-8& 4\\ -1& 5\end{array}|$

9. $|\begin{array}{rr}\hfill 1& \hfill 0\\ \hfill 3& \hfill -4\end{array}|$

10. $|\begin{array}{rr}\hfill 10& \hfill 20\\ \hfill 0& \hfill -10\end{array}|$

11. $|\begin{array}{cc}10& 0.2\\ 5& 0.1\end{array}|$

12. $|\begin{array}{rr}\hfill 6& \hfill -3\\ \hfill 8& \hfill 4\end{array}|$

13. $|\begin{array}{rr}\hfill -2& \hfill -3\\ \hfill 3.1& \hfill 4,000\end{array}|$

14. $|\begin{array}{rr}\hfill -1.1& \hfill 0.6\\ \hfill 7.2& \hfill -0.5\end{array}|$

15. $|\begin{array}{rrr}\hfill -1& \hfill 0& \hfill 0\\ \hfill 0& \hfill 1& \hfill 0\\ \hfill 0& \hfill 0& \hfill -3\end{array}|$

16. $|\begin{array}{rrr}\hfill -1& \hfill 4& \hfill 0\\ \hfill 0& \hfill 2& \hfill 3\\ \hfill 0& \hfill 0& \hfill -3\end{array}|$

17. $|\begin{array}{ccc}1& 0& 1\\ 0& 1& 0\\ 1& 0& 0\end{array}|$

18. $|\begin{array}{rrr}\hfill 2& \hfill -3& \hfill 1\\ \hfill 3& \hfill -4& \hfill 1\\ \hfill -5& \hfill 6& \hfill 1\end{array}|$

19. $|\begin{array}{rrr}\hfill -2& \hfill 1& \hfill 4\\ \hfill -4& \hfill 2& \hfill -8\\ \hfill 2& \hfill -8& \hfill -3\end{array}|$

20. $|\begin{array}{rrr}\hfill 6& \hfill -1& \hfill 2\\ \hfill -4& \hfill -3& \hfill 5\\ \hfill 1& \hfill 9& \hfill -1\end{array}|$

21. $|\begin{array}{rrr}\hfill 5& \hfill 1& \hfill -1\\ \hfill 2& \hfill 3& \hfill 1\\ \hfill 3& \hfill -6& \hfill -3\end{array}|$

22. $|\begin{array}{rrr}\hfill 1.1& \hfill 2& \hfill -1\\ \hfill -4& \hfill 0& \hfill 0\\ \hfill 4.1& \hfill -0.4& \hfill 2.5\end{array}|$

23. $|\begin{array}{rrr}\hfill 2& \hfill -1.6& \hfill 3.1\\ \hfill 1.1& \hfill 3& \hfill -8\\ \hfill -9.3& \hfill 0& \hfill 2\end{array}|$

24. $|\begin{array}{ccc}-\frac{1}{2}& \frac{1}{3}& \frac{1}{4}\\ \frac{1}{5}& -\frac{1}{6}& \frac{1}{7}\\ 0& 0& \frac{1}{8}\end{array}|$

For the following exercises, solve the system of linear equations using Cramer’s Rule.

25. $\begin{array}{l}2x - 3y=-1\\ 4x+5y=9\end{array}$

26. $\begin{array}{r}5x - 4y=2\\ -4x+7y=6\end{array}$

27. $\begin{array}{l}\text{ }6x - 3y=2\hfill \\ -8x+9y=-1\hfill \end{array}$

28. $\begin{array}{l}2x+6y=12\\ 5x - 2y=13\end{array}$

29. $\begin{array}{l}4x+3y=23\hfill \\ \text{ }2x-y=-1\hfill \end{array}$

30. $\begin{array}{l}10x - 6y=2\hfill \\ -5x+8y=-1\hfill \end{array}$

31. $\begin{array}{l}4x - 3y=-3\\ 2x+6y=-4\end{array}$

32. $\begin{array}{r}4x - 5y=7\\ -3x+9y=0\end{array}$

33. $\begin{array}{l}4x+10y=180\hfill \\ -3x - 5y=-105\hfill \end{array}$

34. $\begin{array}{l}\text{ }8x - 2y=-3\hfill \\ -4x+6y=4\hfill \end{array}$

For the following exercises, solve the system of linear equations using Cramer’s Rule.

35. $\begin{array}{l}\text{ }x+2y - 4z=-1\hfill \\ \text{ }7x+3y+5z=26\hfill \\ -2x - 6y+7z=-6\hfill \end{array}$

36. $\begin{array}{l}-5x+2y - 4z=-47\hfill \\ \text{ }4x - 3y-z=-94\hfill \\ \text{ }3x - 3y+2z=94\hfill \end{array}$

37. $\begin{array}{l}\text{ }4x+5y-z=-7\hfill \\ -2x - 9y+2z=8\hfill \\ \text{ }5y+7z=21\hfill \end{array}$

38. $\begin{array}{r}4x - 3y+4z=10\\ 5x - 2z=-2\\ 3x+2y - 5z=-9\end{array}$

39. $\begin{array}{l}4x - 2y+3z=6\hfill \\ \text{ }-6x+y=-2\hfill \\ 2x+7y+8z=24\hfill \end{array}$

40. $\begin{array}{r}\hfill 5x+2y-z=1\\ \hfill -7x - 8y+3z=1.5\\ \hfill 6x - 12y+z=7\end{array}$

41. $\begin{array}{l}\text{ }13x - 17y+16z=73\hfill \\ -11x+15y+17z=61\hfill \\ \text{ }46x+10y - 30z=-18\hfill \end{array}$

42. $\begin{array}{l}\begin{array}{l}\hfill \\ -4x - 3y - 8z=-7\hfill \end{array}\hfill \\ \text{ }2x - 9y+5z=0.5\hfill \\ \text{ }5x - 6y - 5z=-2\hfill \end{array}$

43. $\begin{array}{l}\text{ }4x - 6y+8z=10\hfill \\ -2x+3y - 4z=-5\hfill \\ \text{ }x+y+z=1\hfill \end{array}$

44. $\begin{array}{r}\hfill 4x - 6y+8z=10\\ \hfill -2x+3y - 4z=-5\\ \hfill 12x+18y - 24z=-30\end{array}$

For the following exercises, use the determinant function on a graphing utility.

45. $|\begin{array}{rrrr}\hfill 1& \hfill 0& \hfill 8& \hfill 9\\ \hfill 0& \hfill 2& \hfill 1& \hfill 0\\ \hfill 1& \hfill 0& \hfill 3& \hfill 0\\ \hfill 0& \hfill 2& \hfill 4& \hfill 3\end{array}|$

46. $|\begin{array}{rrrr}\hfill 1& \hfill 0& \hfill 2& \hfill 1\\ \hfill 0& \hfill -9& \hfill 1& \hfill 3\\ \hfill 3& \hfill 0& \hfill -2& \hfill -1\\ \hfill 0& \hfill 1& \hfill 1& \hfill -2\end{array}|$

47. $|\begin{array}{rrrr}\hfill \frac{1}{2}& \hfill 1& \hfill 7& \hfill 4\\ \hfill 0& \hfill \frac{1}{2}& \hfill 100& \hfill 5\\ \hfill 0& \hfill 0& \hfill 2& \hfill 2,000\\ \hfill 0& \hfill 0& \hfill 0& \hfill 2\end{array}|$

48. $|\begin{array}{rrrr}\hfill 1& \hfill 0& \hfill 0& \hfill 0\\ \hfill 2& \hfill 3& \hfill 0& \hfill 0\\ \hfill 4& \hfill 5& \hfill 6& \hfill 0\\ \hfill 7& \hfill 8& \hfill 9& \hfill 0\end{array}|$

For the following exercises, create a system of linear equations to describe the behavior. Then, calculate the determinant. Will there be a unique solution? If so, find the unique solution.

49. Two numbers add up to 56. One number is 20 less than the other.

50. Two numbers add up to 104. If you add two times the first number plus two times the second number, your total is 208

51. Three numbers add up to 106. The first number is 3 less than the second number. The third number is 4 more than the first number.

52. Three numbers add to 216. The sum of the first two numbers is 112. The third number is 8 less than the first two numbers combined.

For the following exercises, create a system of linear equations to describe the behavior. Then, solve the system for all solutions using Cramer’s Rule.

53. You invest $10,000 into two accounts, which receive 8% interest and 5% interest. At the end of a year, you had$10,710 in your combined accounts. How much was invested in each account?

54. You invest $80,000 into two accounts,$22,000 in one account, and $58,000 in the other account. At the end of one year, assuming simple interest, you have earned$2,470 in interest. The second account receives half a percent less than twice the interest on the first account. What are the interest rates for your accounts?

55. A movie theater needs to know how many adult tickets and children tickets were sold out of the 1,200 total tickets. If children’s tickets are $5.95, adult tickets are$11.15, and the total amount of revenue was $12,756, how many children’s tickets and adult tickets were sold? 56. A concert venue sells single tickets for$40 each and couple’s tickets for $65. If the total revenue was$18,090 and the 321 tickets were sold, how many single tickets and how many couple’s tickets were sold?

57. You decide to paint your kitchen green. You create the color of paint by mixing yellow and blue paints. You cannot remember how many gallons of each color went into your mix, but you know there were 10 gal total. Additionally, you kept your receipt, and know the total amount spent was $29.50. If each gallon of yellow costs$2.59, and each gallon of blue costs $3.19, how many gallons of each color go into your green mix? 58. You sold two types of scarves at a farmers’ market and would like to know which one was more popular. The total number of scarves sold was 56, the yellow scarf cost$10, and the purple scarf cost $11. If you had total revenue of$583, how many yellow scarves and how many purple scarves were sold?

59. Your garden produced two types of tomatoes, one green and one red. The red weigh 10 oz, and the green weigh 4 oz. You have 30 tomatoes, and a total weight of 13 lb, 14 oz. How many of each type of tomato do you have?

60. At a market, the three most popular vegetables make up 53% of vegetable sales. Corn has 4% higher sales than broccoli, which has 5% more sales than onions. What percentage does each vegetable have in the market share?

61. At the same market, the three most popular fruits make up 37% of the total fruit sold. Strawberries sell twice as much as oranges, and kiwis sell one more percentage point than oranges. For each fruit, find the percentage of total fruit sold.

62. Three bands performed at a concert venue. The first band charged $15 per ticket, the second band charged$45 per ticket, and the final band charged $22 per ticket. There were 510 tickets sold, for a total of$12,700. If the first band had 40 more audience members than the second band, how many tickets were sold for each band?

63. A movie theatre sold tickets to three movies. The tickets to the first movie were $5, the tickets to the second movie were$11, and the third movie was $12. 100 tickets were sold to the first movie. The total number of tickets sold was 642, for a total revenue of$6,774. How many tickets for each movie were sold?

64. Men aged 20–29, 30–39, and 40–49 made up 78% of the population at a prison last year. This year, the same age groups made up 82.08% of the population. The 20–29 age group increased by 20%, the 30–39 age group increased by 2%, and the 40–49 age group decreased to $\frac{3}{4}$ of their previous population. Originally, the 30–39 age group had 2% more prisoners than the 20–29 age group. Determine the prison population percentage for each age group last year.

65. At a women’s prison down the road, the total number of inmates aged 20–49 totaled 5,525. This year, the 20–29 age group increased by 10%, the 30–39 age group decreased by 20%, and the 40–49 age group doubled. There are now 6,040 prisoners. Originally, there were 500 more in the 30–39 age group than the 20–29 age group. Determine the prison population for each age group last year.

For the following exercises, use this scenario: A health-conscious company decides to make a trail mix out of almonds, dried cranberries, and chocolate-covered cashews. The nutritional information for these items is shown below.

Fat (g) Protein (g) Carbohydrates (g)
Almonds (10) 6 2 3
Cranberries (10) 0.02 0 8
Cashews (10) 7 3.5 5.5

66. For the special “low-carb”trail mix, there are 1,000 pieces of mix. The total number of carbohydrates is 425 g, and the total amount of fat is 570.2 g. If there are 200 more pieces of cashews than cranberries, how many of each item is in the trail mix?

67. For the “hiking” mix, there are 1,000 pieces in the mix, containing 390.8 g of fat, and 165 g of protein. If there is the same amount of almonds as cashews, how many of each item is in the trail mix?

68. For the “energy-booster” mix, there are 1,000 pieces in the mix, containing 145 g of protein and 625 g of carbohydrates. If the number of almonds and cashews summed together is equivalent to the amount of cranberries, how many of each item is in the trail mix?