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 [latex]2\times 2[/latex] matrix.

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

4. The determinant of [latex]2\times 2[/latex] matrix [latex]A[/latex] 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. [latex]|\begin{array}{cc}1& 2\\ 3& 4\end{array}|[/latex]

6. [latex]|\begin{array}{rr}\hfill -1& \hfill 2\\ \hfill 3& \hfill -4\end{array}|[/latex]

7. [latex]|\begin{array}{rr}\hfill 2& \hfill -5\\ \hfill -1& \hfill 6\end{array}|[/latex]

8. [latex]|\begin{array}{cc}-8& 4\\ -1& 5\end{array}|[/latex]

9. [latex]|\begin{array}{rr}\hfill 1& \hfill 0\\ \hfill 3& \hfill -4\end{array}|[/latex]

10. [latex]|\begin{array}{rr}\hfill 10& \hfill 20\\ \hfill 0& \hfill -10\end{array}|[/latex]

11. [latex]|\begin{array}{cc}10& 0.2\\ 5& 0.1\end{array}|[/latex]

12. [latex]|\begin{array}{rr}\hfill 6& \hfill -3\\ \hfill 8& \hfill 4\end{array}|[/latex]

13. [latex]|\begin{array}{rr}\hfill -2& \hfill -3\\ \hfill 3.1& \hfill 4,000\end{array}|[/latex]

14. [latex]|\begin{array}{rr}\hfill -1.1& \hfill 0.6\\ \hfill 7.2& \hfill -0.5\end{array}|[/latex]

15. [latex]|\begin{array}{rrr}\hfill -1& \hfill 0& \hfill 0\\ \hfill 0& \hfill 1& \hfill 0\\ \hfill 0& \hfill 0& \hfill -3\end{array}|[/latex]

16. [latex]|\begin{array}{rrr}\hfill -1& \hfill 4& \hfill 0\\ \hfill 0& \hfill 2& \hfill 3\\ \hfill 0& \hfill 0& \hfill -3\end{array}|[/latex]

17. [latex]|\begin{array}{ccc}1& 0& 1\\ 0& 1& 0\\ 1& 0& 0\end{array}|[/latex]

18. [latex]|\begin{array}{rrr}\hfill 2& \hfill -3& \hfill 1\\ \hfill 3& \hfill -4& \hfill 1\\ \hfill -5& \hfill 6& \hfill 1\end{array}|[/latex]

19. [latex]|\begin{array}{rrr}\hfill -2& \hfill 1& \hfill 4\\ \hfill -4& \hfill 2& \hfill -8\\ \hfill 2& \hfill -8& \hfill -3\end{array}|[/latex]

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

21. [latex]|\begin{array}{rrr}\hfill 5& \hfill 1& \hfill -1\\ \hfill 2& \hfill 3& \hfill 1\\ \hfill 3& \hfill -6& \hfill -3\end{array}|[/latex]

22. [latex]|\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}|[/latex]

23. [latex]|\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}|[/latex]

24. [latex]|\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}|[/latex]

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

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

26. [latex]\begin{array}{r}5x - 4y=2\\ -4x+7y=6\end{array}[/latex]

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

28. [latex]\begin{array}{l}2x+6y=12\\ 5x - 2y=13\end{array}[/latex]

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

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

31. [latex]\begin{array}{l}4x - 3y=-3\\ 2x+6y=-4\end{array}[/latex]

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

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

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

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

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

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

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

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

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

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

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

42. [latex]\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}[/latex]

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

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

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

45. [latex]|\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}|[/latex]

46. [latex]|\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}|[/latex]

47. [latex]|\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}|[/latex]

48. [latex]|\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}|[/latex]