Mathematics 10 — Exercise Review · Q3 (iii)

Full question

Verify (A+B)C=AC+BC.

Step-by-step solution

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

2. The orders match. Combine only entries in the same row and column.

   $$(\left[\begin{array}{cc}2& 3\\ 4& 5\end{array}\right],\left[\begin{array}{cc}5& 6\\ 2& 1\end{array}\right])$$

3. Entry (1,1): add the corresponding entries.

   $$2+5=7$$

4. Entry (1,2): add the corresponding entries.

   $$3+6=9$$

5. Entry (2,1): add the corresponding entries.

   $$4+2=6$$

6. Entry (2,2): add the corresponding entries.

   $$5+1=6$$

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

   $$\left[\begin{array}{cc}7& 9\\ 6& 6\end{array}\right]$$

8. 2 columns in the first matrix match 2 rows in the second. The product has 2 rows and 2 columns.

   $$(\left[\begin{array}{cc}7& 9\\ 6& 6\end{array}\right],\left[\begin{array}{cc}3& 2\\ 1& 7\end{array}\right])$$

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

   $$7\times 3+9\times 1$$

10. Calculate this individual product, including its signs.

    $$7\times 3=21$$

11. Calculate this individual product, including its signs.

    $$9\times 1=9$$

12. Add the calculated products to obtain this entry.

    $$21+9=30$$

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

    $$7\times 2+9\times 7$$

14. Calculate this individual product, including its signs.

    $$7\times 2=14$$

15. Calculate this individual product, including its signs.

    $$9\times 7=63$$

16. Add the calculated products to obtain this entry.

    $$14+63=77$$

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

    $$6\times 3+6\times 1$$

18. Calculate this individual product, including its signs.

    $$6\times 3=18$$

19. Calculate this individual product, including its signs.

    $$6\times 1=6$$

20. Add the calculated products to obtain this entry.

    $$18+6=24$$

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

    $$6\times 2+6\times 7$$

22. Calculate this individual product, including its signs.

    $$6\times 2=12$$

23. Calculate this individual product, including its signs.

    $$6\times 7=42$$

24. Add the calculated products to obtain this entry.

    $$12+42=54$$

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

    $$\left[\begin{array}{cc}30& 77\\ 24& 54\end{array}\right]$$

26. Now compute the right side independently.

27. 2 columns in the first matrix match 2 rows in the second. The product has 2 rows and 2 columns.

    $$(\left[\begin{array}{cc}2& 3\\ 4& 5\end{array}\right],\left[\begin{array}{cc}3& 2\\ 1& 7\end{array}\right])$$

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

    $$2\times 3+3\times 1$$

29. Calculate this individual product, including its signs.

    $$2\times 3=6$$

30. Calculate this individual product, including its signs.

    $$3\times 1=3$$

31. Add the calculated products to obtain this entry.

    $$6+3=9$$

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

    $$2\times 2+3\times 7$$

33. Calculate this individual product, including its signs.

    $$2\times 2=4$$

34. Calculate this individual product, including its signs.

    $$3\times 7=21$$

35. Add the calculated products to obtain this entry.

    $$4+21=25$$

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

    $$4\times 3+5\times 1$$

37. Calculate this individual product, including its signs.

    $$4\times 3=12$$

38. Calculate this individual product, including its signs.

    $$5\times 1=5$$

39. Add the calculated products to obtain this entry.

    $$12+5=17$$

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

    $$4\times 2+5\times 7$$

41. Calculate this individual product, including its signs.

    $$4\times 2=8$$

42. Calculate this individual product, including its signs.

    $$5\times 7=35$$

43. Add the calculated products to obtain this entry.

    $$8+35=43$$

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

    $$\left[\begin{array}{cc}9& 25\\ 17& 43\end{array}\right]$$

45. 2 columns in the first matrix match 2 rows in the second. The product has 2 rows and 2 columns.

    $$(\left[\begin{array}{cc}5& 6\\ 2& 1\end{array}\right],\left[\begin{array}{cc}3& 2\\ 1& 7\end{array}\right])$$

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

    $$5\times 3+6\times 1$$

47. Calculate this individual product, including its signs.

    $$5\times 3=15$$

48. Calculate this individual product, including its signs.

    $$6\times 1=6$$

49. Add the calculated products to obtain this entry.

    $$15+6=21$$

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

    $$5\times 2+6\times 7$$

51. Calculate this individual product, including its signs.

    $$5\times 2=10$$

52. Calculate this individual product, including its signs.

    $$6\times 7=42$$

53. Add the calculated products to obtain this entry.

    $$10+42=52$$

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

    $$2\times 3+1\times 1$$

55. Calculate this individual product, including its signs.

    $$2\times 3=6$$

56. Calculate this individual product, including its signs.

    $$1\times 1=1$$

57. Add the calculated products to obtain this entry.

    $$6+1=7$$

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

    $$2\times 2+1\times 7$$

59. Calculate this individual product, including its signs.

    $$2\times 2=4$$

60. Calculate this individual product, including its signs.

    $$1\times 7=7$$

61. Add the calculated products to obtain this entry.

    $$4+7=11$$

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

    $$\left[\begin{array}{cc}21& 52\\ 7& 11\end{array}\right]$$

63. The orders match. Combine only entries in the same row and column.

    $$(\left[\begin{array}{cc}9& 25\\ 17& 43\end{array}\right],\left[\begin{array}{cc}21& 52\\ 7& 11\end{array}\right])$$

64. Entry (1,1): add the corresponding entries.

    $$9+21=30$$

65. Entry (1,2): add the corresponding entries.

    $$25+52=77$$

66. Entry (2,1): add the corresponding entries.

    $$17+7=24$$

67. Entry (2,2): add the corresponding entries.

    $$43+11=54$$

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

    $$\left[\begin{array}{cc}30& 77\\ 24& 54\end{array}\right]$$

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

    $$(\left[\begin{array}{cc}30& 77\\ 24& 54\end{array}\right],\left[\begin{array}{cc}30& 77\\ 24& 54\end{array}\right])$$

Answer

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