Problem Set 31: Matrices and Matrix Operations

1. Can we add any two matrices together? If so, explain why; if not, explain why not and give an example of two matrices that cannot be added together.

2. Can we multiply any column matrix by any row matrix? Explain why or why not.

3. Can both the products [latex]AB[/latex] and [latex]BA[/latex] be defined? If so, explain how; if not, explain why.

4. Can any two matrices of the same size be multiplied? If so, explain why, and if not, explain why not and give an example of two matrices of the same size that cannot be multiplied together.

5. Does matrix multiplication commute? That is, does [latex]AB=BA?[/latex] If so, prove why it does. If not, explain why it does not.

For the following exercises, use the matrices below and perform the matrix addition or subtraction. Indicate if the operation is undefined.

[latex]A=\left[\begin{array}{cc}1& 3\\ 0& 7\end{array}\right],B=\left[\begin{array}{cc}2& 14\\ 22& 6\end{array}\right],C=\left[\begin{array}{cc}1& 5\\ 8& 92\\ 12& 6\end{array}\right],D=\left[\begin{array}{cc}10& 14\\ 7& 2\\ 5& 61\end{array}\right],E=\left[\begin{array}{cc}6& 12\\ 14& 5\end{array}\right],F=\left[\begin{array}{cc}0& 9\\ 78& 17\\ 15& 4\end{array}\right][/latex]

6. [latex]A+B[/latex]

7. [latex]C+D[/latex]

8. [latex]A+C[/latex]

9. [latex]B-E[/latex]

10. [latex]C+F[/latex]

11. [latex]D-B[/latex]

For the following exercises, use the matrices below to perform scalar multiplication.

[latex]A=\left[\begin{array}{rr}\hfill 4& \hfill 6\\ \hfill 13& \hfill 12\end{array}\right],B=\left[\begin{array}{rr}\hfill 3& \hfill 9\\ \hfill 21& \hfill 12\\ \hfill 0& \hfill 64\end{array}\right],C=\left[\begin{array}{rrrr}\hfill 16& \hfill 3& \hfill 7& \hfill 18\\ \hfill 90& \hfill 5& \hfill 3& \hfill 29\end{array}\right],D=\left[\begin{array}{rrr}\hfill 18& \hfill 12& \hfill 13\\ \hfill 8& \hfill 14& \hfill 6\\ \hfill 7& \hfill 4& \hfill 21\end{array}\right][/latex]

12. [latex]5A[/latex]

13. [latex]3B[/latex]

14. [latex]-2B[/latex]

15. [latex]-4C[/latex]

16. [latex]\frac{1}{2}C[/latex]

17. [latex]100D[/latex]

For the following exercises, use the matrices below to perform matrix multiplication.

[latex]A=\left[\begin{array}{rr}\hfill -1& \hfill 5\\ \hfill 3& \hfill 2\end{array}\right],B=\left[\begin{array}{rrr}\hfill 3& \hfill 6& \hfill 4\\ \hfill -8& \hfill 0& \hfill 12\end{array}\right],C=\left[\begin{array}{rr}\hfill 4& \hfill 10\\ \hfill -2& \hfill 6\\ \hfill 5& \hfill 9\end{array}\right],D=\left[\begin{array}{rrr}\hfill 2& \hfill -3& \hfill 12\\ \hfill 9& \hfill 3& \hfill 1\\ \hfill 0& \hfill 8& \hfill -10\end{array}\right][/latex]

18. [latex]AB[/latex]

19. [latex]BC[/latex]

20. [latex]CA[/latex]

21. [latex]BD[/latex]

22. [latex]DC[/latex]

23. [latex]CB[/latex]

For the following exercises, use the matrices below to perform the indicated operation if possible. If not possible, explain why the operation cannot be performed.

[latex]A=\left[\begin{array}{rr}\hfill 2& \hfill -5\\ \hfill 6& \hfill 7\end{array}\right],B=\left[\begin{array}{rr}\hfill -9& \hfill 6\\ \hfill -4& \hfill 2\end{array}\right],C=\left[\begin{array}{rr}\hfill 0& \hfill 9\\ \hfill 7& \hfill 1\end{array}\right],D=\left[\begin{array}{rrr}\hfill -8& \hfill 7& \hfill -5\\ \hfill 4& \hfill 3& \hfill 2\\ \hfill 0& \hfill 9& \hfill 2\end{array}\right],E=\left[\begin{array}{rrr}\hfill 4& \hfill 5& \hfill 3\\ \hfill 7& \hfill -6& \hfill -5\\ \hfill 1& \hfill 0& \hfill 9\end{array}\right][/latex]

24. [latex]A+B-C[/latex]

25. [latex]4A+5D[/latex]

26. [latex]2C+B[/latex]

27. [latex]3D+4E[/latex]

28. [latex]C - 0.5D[/latex]

29. [latex]100D - 10E[/latex]

For the following exercises, use the matrices below to perform the indicated operation if possible. If not possible, explain why the operation cannot be performed. (Hint: [latex]{A}^{2}=A\cdot A[/latex])

[latex]A=\left[\begin{array}{rr}\hfill -10& \hfill 20\\ \hfill 5& \hfill 25\end{array}\right],B=\left[\begin{array}{rr}\hfill 40& \hfill 10\\ \hfill -20& \hfill 30\end{array}\right],C=\left[\begin{array}{rr}\hfill -1& \hfill 0\\ \hfill 0& \hfill -1\\ \hfill 1& \hfill 0\end{array}\right][/latex]

30. [latex]AB[/latex]

31. [latex]BA[/latex]

32. [latex]CA[/latex]

33. [latex]BC[/latex]

34. [latex]{A}^{2}[/latex]

35. [latex]{B}^{2}[/latex]

36. [latex]{C}^{2}[/latex]

37. [latex]{B}^{2}{A}^{2}[/latex]

38. [latex]{A}^{2}{B}^{2}[/latex]

39. [latex]{\left(AB\right)}^{2}[/latex]

40. [latex]{\left(BA\right)}^{2}[/latex]

For the following exercises, use the matrices below to perform the indicated operation if possible. If not possible, explain why the operation cannot be performed. (Hint: [latex]{A}^{2}=A\cdot A[/latex])

[latex]A=\left[\begin{array}{rr}\hfill 1& \hfill 0\\ \hfill 2& \hfill 3\end{array}\right],B=\left[\begin{array}{rrr}\hfill -2& \hfill 3& \hfill 4\\ \hfill -1& \hfill 1& \hfill -5\end{array}\right],C=\left[\begin{array}{rr}\hfill 0.5& \hfill 0.1\\ \hfill 1& \hfill 0.2\\ \hfill -0.5& \hfill 0.3\end{array}\right],D=\left[\begin{array}{rrr}\hfill 1& \hfill 0& \hfill -1\\ \hfill -6& \hfill 7& \hfill 5\\ \hfill 4& \hfill 2& \hfill 1\end{array}\right][/latex]

41. [latex]AB[/latex]

42. [latex]BA[/latex]

43. [latex]BD[/latex]

44. [latex]DC[/latex]

45. [latex]{D}^{2}[/latex]

46. [latex]{A}^{2}[/latex]

47. [latex]{D}^{3}[/latex]

48. [latex]\left(AB\right)C[/latex]

49. [latex]A\left(BC\right)[/latex]

For the following exercises, use the matrices below to perform the indicated operation if possible. If not possible, explain why the operation cannot be performed. Use a calculator to verify your solution.

[latex]A=\left[\begin{array}{rrr}\hfill -2& \hfill 0& \hfill 9\\ \hfill 1& \hfill 8& \hfill -3\\ \hfill 0.5& \hfill 4& \hfill 5\end{array}\right],B=\left[\begin{array}{rrr}\hfill 0.5& \hfill 3& \hfill 0\\ \hfill -4& \hfill 1& \hfill 6\\ \hfill 8& \hfill 7& \hfill 2\end{array}\right],C=\left[\begin{array}{rrr}\hfill 1& \hfill 0& \hfill 1\\ \hfill 0& \hfill 1& \hfill 0\\ \hfill 1& \hfill 0& \hfill 1\end{array}\right][/latex]

50. [latex]AB[/latex]

51. [latex]BA[/latex]

52. [latex]CA[/latex]

53. [latex]BC[/latex]

54. [latex]ABC[/latex]

For the following exercises, use the matrix below to perform the indicated operation on the given matrix.

[latex]B=\left[\begin{array}{rrr}\hfill 1& \hfill 0& \hfill 0\\ \hfill 0& \hfill 0& \hfill 1\\ \hfill 0& \hfill 1& \hfill 0\end{array}\right][/latex]

55. [latex]{B}^{2}[/latex]

56. [latex]{B}^{3}[/latex]

57. [latex]{B}^{4}[/latex]

58. [latex]{B}^{5}[/latex]

59. Using the above questions, find a formula for [latex]{B}^{n}[/latex]. Test the formula for [latex]{B}^{201}[/latex] and [latex]{B}^{202},\text{}[/latex] using a calculator.