Mathematics 10 — Exercise 3.4 · Q2 (ii)

Full question

Verify A(B−C) = AB−AC.

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.

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

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

   $$2-5=-3$$

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

   $$-1-1=-2$$

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

   $$0-1=-1$$

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

   $$3-4=-1$$

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

   $$\left[\begin{array}{cc}-3& -2\\ -1& -1\end{array}\right]$$

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

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

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

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

10. Calculate this individual product, including its signs.

    $$5\times \left(-3\right)=-15$$

11. Calculate this individual product, including its signs.

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

12. Add the calculated products to obtain this entry.

    $$-15-1=-16$$

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

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

14. Calculate this individual product, including its signs.

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

15. Calculate this individual product, including its signs.

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

16. Add the calculated products to obtain this entry.

    $$-10-1=-11$$

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

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

18. Calculate this individual product, including its signs.

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

19. Calculate this individual product, including its signs.

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

20. Add the calculated products to obtain this entry.

    $$3-4=-1$$

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

    $$--2+4\times \left(-1\right)$$

22. Calculate this individual product, including its signs.

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

23. Calculate this individual product, including its signs.

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

24. Add the calculated products to obtain this entry.

    $$2-4=-2$$

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

    $$\left[\begin{array}{cc}-16& -11\\ -1& -2\end{array}\right]$$

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.

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

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

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

29. Calculate this individual product, including its signs.

    $$5\times 2=10$$

30. Calculate this individual product, including its signs.

    $$1\times 0=0$$

31. Add the calculated products to obtain this entry.

    $$10+0=10$$

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

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

33. Calculate this individual product, including its signs.

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

34. Calculate this individual product, including its signs.

    $$1\times 3=3$$

35. Add the calculated products to obtain this entry.

    $$-5+3=-2$$

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

    $$-2+4\times 0$$

37. Calculate this individual product, including its signs.

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

38. Calculate this individual product, including its signs.

    $$4\times 0=0$$

39. Add the calculated products to obtain this entry.

    $$-2+0=-2$$

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

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

41. Calculate this individual product, including its signs.

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

42. Calculate this individual product, including its signs.

    $$4\times 3=12$$

43. Add the calculated products to obtain this entry.

    $$1+12=13$$

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

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

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

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

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

    $$5\times 5+1\times 1$$

47. Calculate this individual product, including its signs.

    $$5\times 5=25$$

48. Calculate this individual product, including its signs.

    $$1\times 1=1$$

49. Add the calculated products to obtain this entry.

    $$25+1=26$$

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

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

51. Calculate this individual product, including its signs.

    $$5\times 1=5$$

52. Calculate this individual product, including its signs.

    $$1\times 4=4$$

53. Add the calculated products to obtain this entry.

    $$5+4=9$$

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

    $$-5+4\times 1$$

55. Calculate this individual product, including its signs.

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

56. Calculate this individual product, including its signs.

    $$4\times 1=4$$

57. Add the calculated products to obtain this entry.

    $$-5+4=-1$$

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

    $$-1+4\times 4$$

59. Calculate this individual product, including its signs.

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

60. Calculate this individual product, including its signs.

    $$4\times 4=16$$

61. Add the calculated products to obtain this entry.

    $$-1+16=15$$

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

    $$\left[\begin{array}{cc}26& 9\\ -1& 15\end{array}\right]$$

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

    $$(\left[\begin{array}{cc}10& -2\\ -2& 13\end{array}\right],\left[\begin{array}{cc}26& 9\\ -1& 15\end{array}\right])$$

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

    $$10-26=-16$$

65. Calculate this operation before combining the remaining terms.

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

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

    $$-2-9=-11$$

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

    $$-2+1=-1$$

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

    $$13-15=-2$$

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

    $$\left[\begin{array}{cc}-16& -11\\ -1& -2\end{array}\right]$$

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

    $$(\left[\begin{array}{cc}-16& -11\\ -1& -2\end{array}\right],\left[\begin{array}{cc}-16& -11\\ -1& -2\end{array}\right])$$

Answer

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