Mathematics 10 — Exercise 3.4 · Q2 (v)

Full question

Verify (B+C)A = BA+CA.

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): add the corresponding entries.

   $$2+5=7$$

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

   $$-1+1=0$$

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

   $$0+1=1$$

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

   $$3+4=7$$

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

   $$\left[\begin{array}{cc}7& 0\\ 1& 7\end{array}\right]$$

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

   $$(\left[\begin{array}{cc}7& 0\\ 1& 7\end{array}\right],\left[\begin{array}{cc}5& 1\\ -1& 4\end{array}\right])$$

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

   $$7\times 5+0\times \left(-1\right)$$

10. Calculate this individual product, including its signs.

    $$7\times 5=35$$

11. Calculate this individual product, including its signs.

    $$0\times \left(-1\right)=0$$

12. Add the calculated products to obtain this entry.

    $$35+0=35$$

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

    $$7\times 1+0\times 4$$

14. Calculate this individual product, including its signs.

    $$7\times 1=7$$

15. Calculate this individual product, including its signs.

    $$0\times 4=0$$

16. Add the calculated products to obtain this entry.

    $$7+0=7$$

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

    $$1\times 5+7\times \left(-1\right)$$

18. Calculate this individual product, including its signs.

    $$1\times 5=5$$

19. Calculate this individual product, including its signs.

    $$7\times \left(-1\right)=-7$$

20. Add the calculated products to obtain this entry.

    $$5-7=-2$$

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

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

22. Calculate this individual product, including its signs.

    $$1\times 1=1$$

23. Calculate this individual product, including its signs.

    $$7\times 4=28$$

24. Add the calculated products to obtain this entry.

    $$1+28=29$$

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

    $$\left[\begin{array}{cc}35& 7\\ -2& 29\end{array}\right]$$

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.

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

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

    $$2\times 5--1$$

29. Calculate this individual product, including its signs.

    $$2\times 5=10$$

30. Calculate this individual product, including its signs.

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

31. Add the calculated products to obtain this entry.

    $$10+1=11$$

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

    $$2\times 1-4$$

33. Calculate this individual product, including its signs.

    $$2\times 1=2$$

34. Calculate this individual product, including its signs.

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

35. Add the calculated products to obtain this entry.

    $$2-4=-2$$

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

    $$0\times 5+3\times \left(-1\right)$$

37. Calculate this individual product, including its signs.

    $$0\times 5=0$$

38. Calculate this individual product, including its signs.

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

39. Add the calculated products to obtain this entry.

    $$0-3=-3$$

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

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

41. Calculate this individual product, including its signs.

    $$0\times 1=0$$

42. Calculate this individual product, including its signs.

    $$3\times 4=12$$

43. Add the calculated products to obtain this entry.

    $$0+12=12$$

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

    $$\left[\begin{array}{cc}11& -2\\ -3& 12\end{array}\right]$$

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.

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

47. Calculate this individual product, including its signs.

    $$5\times 5=25$$

48. Calculate this individual product, including its signs.

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

49. Add the calculated products to obtain this entry.

    $$25-1=24$$

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.

    $$1\times 5+4\times \left(-1\right)$$

55. Calculate this individual product, including its signs.

    $$1\times 5=5$$

56. Calculate this individual product, including its signs.

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

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.

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

59. Calculate this individual product, including its signs.

    $$1\times 1=1$$

60. Calculate this individual product, including its signs.

    $$4\times 4=16$$

61. Add the calculated products to obtain this entry.

    $$1+16=17$$

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

    $$\left[\begin{array}{cc}24& 9\\ 1& 17\end{array}\right]$$

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

    $$(\left[\begin{array}{cc}11& -2\\ -3& 12\end{array}\right],\left[\begin{array}{cc}24& 9\\ 1& 17\end{array}\right])$$

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

    $$11+24=35$$

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

    $$-2+9=7$$

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

    $$-3+1=-2$$

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

    $$12+17=29$$

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

    $$\left[\begin{array}{cc}35& 7\\ -2& 29\end{array}\right]$$

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

    $$(\left[\begin{array}{cc}35& 7\\ -2& 29\end{array}\right],\left[\begin{array}{cc}35& 7\\ -2& 29\end{array}\right])$$

Answer

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