Mathematics 10 — Exercise 3.4 · Q2 (iii)

Full question

Verify A(BC) = (AB)C.

Step-by-step solution

1. Compute the left side first, following the bracket order.

2. 2 columns in the first matrix match 2 rows in the second. The product has 2 rows and 2 columns.

   $$(\left[\begin{array}{cc}2& -1\\ 0& 3\end{array}\right],\left[\begin{array}{cc}5& 1\\ 1& 4\end{array}\right])$$

3. Entry (1,1): pair row 1 with column 1 and add their products.

   $$2\times 5-1$$

4. Calculate this individual product, including its signs.

   $$2\times 5=10$$

5. Calculate this individual product, including its signs.

   $$\left(-1\right)\times 1=-1$$

6. Add the calculated products to obtain this entry.

   $$10-1=9$$

7. Entry (1,2): pair row 1 with column 2 and add their products.

   $$2\times 1-4$$

8. Calculate this individual product, including its signs.

   $$2\times 1=2$$

9. Calculate this individual product, including its signs.

   $$\left(-1\right)\times 4=-4$$

10. Add the calculated products to obtain this entry.

    $$2-4=-2$$

11. Entry (2,1): pair row 2 with column 1 and add their products.

    $$0\times 5+3\times 1$$

12. Calculate this individual product, including its signs.

    $$0\times 5=0$$

13. Calculate this individual product, including its signs.

    $$3\times 1=3$$

14. Add the calculated products to obtain this entry.

    $$0+3=3$$

15. Entry (2,2): pair row 2 with column 2 and add their products.

    $$0\times 1+3\times 4$$

16. Calculate this individual product, including its signs.

    $$0\times 1=0$$

17. Calculate this individual product, including its signs.

    $$3\times 4=12$$

18. Add the calculated products to obtain this entry.

    $$0+12=12$$

19. Place the calculated entries in their original row and column positions.

    $$\left[\begin{array}{cc}9& -2\\ 3& 12\end{array}\right]$$

20. 2 columns in the first matrix match 2 rows in the second. The product has 2 rows and 2 columns.

    $$(\left[\begin{array}{cc}5& 1\\ -1& 4\end{array}\right],\left[\begin{array}{cc}9& -2\\ 3& 12\end{array}\right])$$

21. Entry (1,1): pair row 1 with column 1 and add their products.

    $$5\times 9+1\times 3$$

22. Calculate this individual product, including its signs.

    $$5\times 9=45$$

23. Calculate this individual product, including its signs.

    $$1\times 3=3$$

24. Add the calculated products to obtain this entry.

    $$45+3=48$$

25. Entry (1,2): pair row 1 with column 2 and add their products.

    $$5\times \left(-2\right)+1\times 12$$

26. Calculate this individual product, including its signs.

    $$5\times \left(-2\right)=-10$$

27. Calculate this individual product, including its signs.

    $$1\times 12=12$$

28. Add the calculated products to obtain this entry.

    $$-10+12=2$$

29. Entry (2,1): pair row 2 with column 1 and add their products.

    $$-9+4\times 3$$

30. Calculate this individual product, including its signs.

    $$\left(-1\right)\times 9=-9$$

31. Calculate this individual product, including its signs.

    $$4\times 3=12$$

32. Add the calculated products to obtain this entry.

    $$-9+12=3$$

33. Entry (2,2): pair row 2 with column 2 and add their products.

    $$--2+4\times 12$$

34. Calculate this individual product, including its signs.

    $$\left(-1\right)\times \left(-2\right)=2$$

35. Calculate this individual product, including its signs.

    $$4\times 12=48$$

36. Add the calculated products to obtain this entry.

    $$2+48=50$$

37. Place the calculated entries in their original row and column positions.

    $$\left[\begin{array}{cc}48& 2\\ 3& 50\end{array}\right]$$

38. Now compute the right side independently.

39. 2 columns in the first matrix match 2 rows in the second. The product has 2 rows and 2 columns.

    $$(\left[\begin{array}{cc}5& 1\\ -1& 4\end{array}\right],\left[\begin{array}{cc}2& -1\\ 0& 3\end{array}\right])$$

40. Entry (1,1): pair row 1 with column 1 and add their products.

    $$5\times 2+1\times 0$$

41. Calculate this individual product, including its signs.

    $$5\times 2=10$$

42. Calculate this individual product, including its signs.

    $$1\times 0=0$$

43. Add the calculated products to obtain this entry.

    $$10+0=10$$

44. Entry (1,2): pair row 1 with column 2 and add their products.

    $$5\times \left(-1\right)+1\times 3$$

45. Calculate this individual product, including its signs.

    $$5\times \left(-1\right)=-5$$

46. Calculate this individual product, including its signs.

    $$1\times 3=3$$

47. Add the calculated products to obtain this entry.

    $$-5+3=-2$$

48. Entry (2,1): pair row 2 with column 1 and add their products.

    $$-2+4\times 0$$

49. Calculate this individual product, including its signs.

    $$\left(-1\right)\times 2=-2$$

50. Calculate this individual product, including its signs.

    $$4\times 0=0$$

51. Add the calculated products to obtain this entry.

    $$-2+0=-2$$

52. Entry (2,2): pair row 2 with column 2 and add their products.

    $$--1+4\times 3$$

53. Calculate this individual product, including its signs.

    $$\left(-1\right)\times \left(-1\right)=1$$

54. Calculate this individual product, including its signs.

    $$4\times 3=12$$

55. Add the calculated products to obtain this entry.

    $$1+12=13$$

56. Place the calculated entries in their original row and column positions.

    $$\left[\begin{array}{cc}10& -2\\ -2& 13\end{array}\right]$$

57. 2 columns in the first matrix match 2 rows in the second. The product has 2 rows and 2 columns.

    $$(\left[\begin{array}{cc}10& -2\\ -2& 13\end{array}\right],\left[\begin{array}{cc}5& 1\\ 1& 4\end{array}\right])$$

58. Entry (1,1): pair row 1 with column 1 and add their products.

    $$10\times 5-2$$

59. Calculate this individual product, including its signs.

    $$10\times 5=50$$

60. Calculate this individual product, including its signs.

    $$\left(-2\right)\times 1=-2$$

61. Add the calculated products to obtain this entry.

    $$50-2=48$$

62. Calculate this operation before combining the remaining terms.

    $$\left(-1\right)\times 8=-8$$

63. Entry (1,2): pair row 1 with column 2 and add their products.

    $$10\times 1-8$$

64. Calculate this individual product, including its signs.

    $$10\times 1=10$$

65. Calculate this individual product, including its signs.

    $$\left(-2\right)\times 4=-8$$

66. Add the calculated products to obtain this entry.

    $$10-8=2$$

67. Entry (2,1): pair row 2 with column 1 and add their products.

    $$-10+13\times 1$$

68. Calculate this individual product, including its signs.

    $$\left(-2\right)\times 5=-10$$

69. Calculate this individual product, including its signs.

    $$13\times 1=13$$

70. Add the calculated products to obtain this entry.

    $$-10+13=3$$

71. Entry (2,2): pair row 2 with column 2 and add their products.

    $$-2+13\times 4$$

72. Calculate this individual product, including its signs.

    $$\left(-2\right)\times 1=-2$$

73. Calculate this individual product, including its signs.

    $$13\times 4=52$$

74. Add the calculated products to obtain this entry.

    $$-2+52=50$$

75. Place the calculated entries in their original row and column positions.

    $$\left[\begin{array}{cc}48& 2\\ 3& 50\end{array}\right]$$

76. Every corresponding entry is equal, so the stated identity holds.

    $$(\left[\begin{array}{cc}48& 2\\ 3& 50\end{array}\right],\left[\begin{array}{cc}48& 2\\ 3& 50\end{array}\right])$$

Answer

Every corresponding entry is equal, so the stated identity holds.