Mathematics 10 — Exercise Review · Q3 (iv)

Full question

Verify C(A−B)=CA−CB.

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): subtract the corresponding entries.

   $$2-5=-3$$

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

   $$3-6=-3$$

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

   $$4-2=2$$

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

   $$5-1=4$$

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

   $$\left[\begin{array}{cc}-3& -3\\ 2& 4\end{array}\right]$$

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

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

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

   $$3\times \left(-3\right)+2\times 2$$

10. Calculate this individual product, including its signs.

    $$3\times \left(-3\right)=-9$$

11. Calculate this individual product, including its signs.

    $$2\times 2=4$$

12. Add the calculated products to obtain this entry.

    $$-9+4=-5$$

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

    $$3\times \left(-3\right)+2\times 4$$

14. Calculate this individual product, including its signs.

    $$3\times \left(-3\right)=-9$$

15. Calculate this individual product, including its signs.

    $$2\times 4=8$$

16. Add the calculated products to obtain this entry.

    $$-9+8=-1$$

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

    $$1\times \left(-3\right)+7\times 2$$

18. Calculate this individual product, including its signs.

    $$1\times \left(-3\right)=-3$$

19. Calculate this individual product, including its signs.

    $$7\times 2=14$$

20. Add the calculated products to obtain this entry.

    $$-3+14=11$$

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

    $$1\times \left(-3\right)+7\times 4$$

22. Calculate this individual product, including its signs.

    $$1\times \left(-3\right)=-3$$

23. Calculate this individual product, including its signs.

    $$7\times 4=28$$

24. Add the calculated products to obtain this entry.

    $$-3+28=25$$

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

    $$\left[\begin{array}{cc}-5& -1\\ 11& 25\end{array}\right]$$

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.

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

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

    $$3\times 2+2\times 4$$

29. Calculate this individual product, including its signs.

    $$3\times 2=6$$

30. Calculate this individual product, including its signs.

    $$2\times 4=8$$

31. Add the calculated products to obtain this entry.

    $$6+8=14$$

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

    $$3\times 3+2\times 5$$

33. Calculate this individual product, including its signs.

    $$3\times 3=9$$

34. Calculate this individual product, including its signs.

    $$2\times 5=10$$

35. Add the calculated products to obtain this entry.

    $$9+10=19$$

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

    $$1\times 2+7\times 4$$

37. Calculate this individual product, including its signs.

    $$1\times 2=2$$

38. Calculate this individual product, including its signs.

    $$7\times 4=28$$

39. Add the calculated products to obtain this entry.

    $$2+28=30$$

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

    $$1\times 3+7\times 5$$

41. Calculate this individual product, including its signs.

    $$1\times 3=3$$

42. Calculate this individual product, including its signs.

    $$7\times 5=35$$

43. Add the calculated products to obtain this entry.

    $$3+35=38$$

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

    $$\left[\begin{array}{cc}14& 19\\ 30& 38\end{array}\right]$$

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

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

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

    $$3\times 5+2\times 2$$

47. Calculate this individual product, including its signs.

    $$3\times 5=15$$

48. Calculate this individual product, including its signs.

    $$2\times 2=4$$

49. Add the calculated products to obtain this entry.

    $$15+4=19$$

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

    $$3\times 6+2\times 1$$

51. Calculate this individual product, including its signs.

    $$3\times 6=18$$

52. Calculate this individual product, including its signs.

    $$2\times 1=2$$

53. Add the calculated products to obtain this entry.

    $$18+2=20$$

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

    $$1\times 5+7\times 2$$

55. Calculate this individual product, including its signs.

    $$1\times 5=5$$

56. Calculate this individual product, including its signs.

    $$7\times 2=14$$

57. Add the calculated products to obtain this entry.

    $$5+14=19$$

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

    $$1\times 6+7\times 1$$

59. Calculate this individual product, including its signs.

    $$1\times 6=6$$

60. Calculate this individual product, including its signs.

    $$7\times 1=7$$

61. Add the calculated products to obtain this entry.

    $$6+7=13$$

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

    $$\left[\begin{array}{cc}19& 20\\ 19& 13\end{array}\right]$$

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

    $$(\left[\begin{array}{cc}14& 19\\ 30& 38\end{array}\right],\left[\begin{array}{cc}19& 20\\ 19& 13\end{array}\right])$$

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

    $$14-19=-5$$

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

    $$19-20=-1$$

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

    $$30-19=11$$

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

    $$38-13=25$$

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

    $$\left[\begin{array}{cc}-5& -1\\ 11& 25\end{array}\right]$$

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

    $$(\left[\begin{array}{cc}-5& -1\\ 11& 25\end{array}\right],\left[\begin{array}{cc}-5& -1\\ 11& 25\end{array}\right])$$

Answer

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