We've updated our
Privacy Policy effective December 15. Please read our updated Privacy Policy and tap

Study Guides > MTH 163, Precalculus

Section Exercises

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 ABAB and BABA 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 AB=BA?AB=BA? 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.

A=[1307],B=[214226],C=[15892126],D=[101472561],E=[612145],F=[097817154]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]

6. A+BA+B 7. C+DC+D 8. A+CA+C 9. BEB-E 10. C+FC+F 11. DBD-B For the following exercises, use the matrices below to perform scalar multiplication.

A=[461312],B=[392112064],C=[163718905329],D=[18121381467421]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]

12. 5A5A 13. 3B3B 14. 2B-2B 15. 4C-4C 16. 12C\frac{1}{2}C 17. 100D100D For the following exercises, use the matrices below to perform matrix multiplication.

A=[1532],B=[3648012],C=[4102659],D=[23129310810]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]

18. ABAB 19. BCBC 20. CACA 21. BDBD 22. DCDC 23. CBCB 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.

A=[2567],B=[9642],C=[0971],D=[875432092],E=[453765109]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]

24. A+BCA+B-C 25. 4A+5D4A+5D 26. 2C+B2C+B 27. 3D+4E3D+4E 28. C0.5DC - 0.5D 29. 100D10E100D - 10E 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: A2=AA{A}^{2}=A\cdot A)

A=[1020525],B=[40102030],C=[100110]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]

30. ABAB 31. BABA 32. CACA 33. BCBC 34. A2{A}^{2} 35. B2{B}^{2} 36. C2{C}^{2} 37. B2A2{B}^{2}{A}^{2} 38. A2B2{A}^{2}{B}^{2} 39. (AB)2{\left(AB\right)}^{2} 40. (BA)2{\left(BA\right)}^{2} 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: A2=AA{A}^{2}=A\cdot A)

A=[1023],B=[234115],C=[0.50.110.20.50.3],D=[101675421]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]

41. ABAB 42. BABA 43. BDBD 44. DCDC 45. D2{D}^{2} 46. A2{A}^{2} 47. D3{D}^{3} 48. (AB)C\left(AB\right)C 49. A(BC)A\left(BC\right) 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.

A=[2091830.545],B=[0.530416872],C=[101010101]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]

50. ABAB 51. BABA 52. CACA 53. BCBC 54. ABCABC For the following exercises, use the matrix below to perform the indicated operation on the given matrix.

B=[100001010]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]

55. B2{B}^{2} 56. B3{B}^{3} 57. B4{B}^{4} 58. B5{B}^{5} 59. Using the above questions, find a formula for Bn{B}^{n}. Test the formula for B201{B}^{201} and B202,{B}^{202},\text{} using a calculator.

Licenses & Attributions

CC licensed content, Specific attribution

  • Precalculus. Provided by: OpenStax Authored by: OpenStax College. Located at: https://cnx.org/contents/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175:1/Preface. License: CC BY: Attribution.