Problem Set: Solving Systems with Cramer’s Rule

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 2×2 matrix.

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

4. The determinant of 2×2 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. |1234|

6. |−123−4|

7. |2−5−16|

8. |−84−15|

9. |103−4|

10. |10200−10|

11. |100.250.1|

12. |6−384|

13. |−2−33.14,000|

14. |−1.10.67.2−0.5|

15. |−10001000−3|

16. |−14002300−3|

17. |101010100|

18. |2−313−41−561|

19. |−214−42−82−8−3|

20. |6−12−4−3519−1|

21. |51−12313−6−3|

22. |1.12−1−4004.1−0.42.5|

23. |2−1.63.11.13−8−9.302|

24. |−12131415−16170018|

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

25. 2x−3y=−14x+5y=9

26. 5x−4y=2−4x+7y=6

27.  6x−3y=2−8x+9y=−1

28. 2x+6y=125x−2y=13

29. 4x+3y=23 2x−y=−1

30. 10x−6y=2−5x+8y=−1

31. 4x−3y=−32x+6y=−4

32. 4x−5y=7−3x+9y=0

33. 4x+10y=180−3x−5y=−105

34.  8x−2y=−3−4x+6y=4

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

35.  x+2y−4z=−1 7x+3y+5z=26−2x−6y+7z=−6

36. −5x+2y−4z=−47 4x−3y−z=−94 3x−3y+2z=94

37.  4x+5y−z=−7−2x−9y+2z=8 5y+7z=21

38. 4x−3y+4z=105x−2z=−23x+2y−5z=−9

39. 4x−2y+3z=6 −6x+y=−22x+7y+8z=24

40. 5x+2y−z=1−7x−8y+3z=1.56x−12y+z=7

41.  13x−17y+16z=73−11x+15y+17z=61 46x+10y−30z=−18

42. −4x−3y−8z=−7 2x−9y+5z=0.5 5x−6y−5z=−2

43.  4x−6y+8z=10−2x+3y−4z=−5 x+y+z=1

44. 4x−6y+8z=10−2x+3y−4z=−512x+18y−24z=−30

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

45. |1089021010300243|

46. |10210−91330−2−1011−2|

47. |1217401210050022,0000002|

48. |1000230045607890|