Mathematics 10 — Exercise Review · Q5 (ii) inverse

Full question

Solve using matrix inversion.

Step-by-step solution

1. Write the coefficients in A, the unknowns in X, and constants in B, giving AX = B.

   $$(\left[\begin{array}{cc}1& -6\\ 2& 6\end{array}\right],\left[\begin{array}{c}x\\ y\end{array}\right],\left[\begin{array}{c}-15\\ -3\end{array}\right])$$

2. For a 2 × 2 determinant, multiply the main diagonal and subtract the other diagonal product.

   $$D=1\times 6-\left(-6\right)\times 2$$

3. Calculate the first diagonal product.

   $$1\times 6=6$$

4. Calculate the second diagonal product.

   $$\left(-6\right)\times 2=-12$$

5. Subtract the second product, including its sign.

   $$D=6+12$$

6. Simplify the subtraction.

   $$D=18$$

7. For a 2 × 2 determinant, multiply the main diagonal and subtract the other diagonal product.

   $$D=1\times 6-\left(-6\right)\times 2$$

8. Calculate the first diagonal product.

   $$1\times 6=6$$

9. Calculate the second diagonal product.

   $$\left(-6\right)\times 2=-12$$

10. Subtract the second product, including its sign.

    $$D=6+12$$

11. Simplify the subtraction.

    $$D=18$$

12. The determinant is nonzero, so the inverse exists.

    $$18\ne 0$$

13. Swap the two diagonal entries to begin the adjoint.

    $$\left[\begin{array}{cc}6& -6\\ 2& 1\end{array}\right]$$

14. Change the signs of the two off-diagonal entries.

    $$\left[\begin{array}{cc}6& 6\\ -2& 1\end{array}\right]$$

15. Divide adjoint entry (1,1) by the determinant.

    $$\frac{6}{18}=\frac{1}{3}$$

16. Divide adjoint entry (1,2) by the determinant.

    $$\frac{6}{18}=\frac{1}{3}$$

17. For multiplication of fractions, multiply the numerators together and the denominators together.

    $$\frac{\left(-2\right)\times 1}{1\times 18}$$

18. Calculate these two products separately.

    $$-\frac{2}{18}$$

19. The numerator and denominator have this greatest common divisor. Divide BOTH by it.

    $$\mathrm{GCD}=2$$

20. Show the two divisions explicitly.

    $$\frac{-\frac{2}{2}}{\frac{18}{2}}$$

21. Evaluate the numerator division.

    $$-\frac{2}{2}=-1$$

22. Evaluate the denominator division.

    $$\frac{18}{2}=9$$

23. Write the reduced fraction.

    $$-\frac{1}{9}$$

24. Calculate this operation before combining the remaining terms.

    $$\left(-2\right)\times \frac{1}{18}=-\frac{1}{9}$$

25. Divide adjoint entry (2,1) by the determinant.

    $$-\frac{2}{18}=-\frac{1}{9}$$

26. Divide adjoint entry (2,2) by the determinant.

    $$\frac{1}{18}=\frac{1}{18}$$

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

    $$\left[\begin{array}{cc}\frac{1}{3}& \frac{1}{3}\\ -\frac{1}{9}& \frac{1}{18}\end{array}\right]$$

28. Premultiply AX = B by A⁻¹. Since A⁻¹A = I, X = A⁻¹B.

29. 2 columns in the first matrix match 2 rows in the second. The product has 2 rows and 1 columns.

    $$(\left[\begin{array}{cc}\frac{1}{3}& \frac{1}{3}\\ -\frac{1}{9}& \frac{1}{18}\end{array}\right],\left[\begin{array}{c}-15\\ -3\end{array}\right])$$

30. For multiplication of fractions, multiply the numerators together and the denominators together.

    $$\frac{1\times 1}{1\times 3}$$

31. Calculate these two products separately.

    $$\frac{1}{3}$$

32. Calculate this operation before combining the remaining terms.

    $$1\times \frac{1}{3}=\frac{1}{3}$$

33. For multiplication of fractions, multiply the numerators together and the denominators together.

    $$\frac{1\times \left(-15\right)}{3\times 1}$$

34. Calculate these two products separately.

    $$-\frac{15}{3}$$

35. The numerator and denominator have this greatest common divisor. Divide BOTH by it.

    $$\mathrm{GCD}=3$$

36. Show the two divisions explicitly.

    $$\frac{-\frac{15}{3}}{\frac{3}{3}}$$

37. Evaluate the numerator division.

    $$-\frac{15}{3}=-5$$

38. Evaluate the denominator division.

    $$\frac{3}{3}=1$$

39. Write the reduced fraction.

    $$-5\times \frac{1}{1}$$

40. For multiplication of fractions, multiply the numerators together and the denominators together.

    $$\frac{1\times \left(-3\right)}{3\times 1}$$

41. Calculate these two products separately.

    $$-\frac{3}{3}$$

42. Show the two divisions explicitly.

    $$\frac{-\frac{3}{3}}{\frac{3}{3}}$$

43. Evaluate the numerator division.

    $$-\frac{3}{3}=-1$$

44. Write the reduced fraction.

    $$-\frac{1}{1}$$

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

    $$\frac{1}{3}\times \left(-15\right)+\frac{1}{3}\times \left(-3\right)$$

46. Calculate this individual product, including its signs.

    $$\frac{1}{3}\times \left(-15\right)=-5$$

47. Calculate this individual product, including its signs.

    $$\frac{1}{3}\times \left(-3\right)=-1$$

48. Calculate this operation before combining the remaining terms.

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

49. Add the calculated products to obtain this entry.

    $$-5-1=-6$$

50. For multiplication of fractions, multiply the numerators together and the denominators together.

    $$\frac{5\times 1}{1\times 3}$$

51. Calculate these two products separately.

    $$\frac{5}{3}$$

52. Calculate this operation before combining the remaining terms.

    $$5\times \frac{1}{3}=\frac{5}{3}$$

53. For multiplication of fractions, multiply the numerators together and the denominators together.

    $$\frac{1\times 1}{1\times 18}$$

54. Calculate these two products separately.

    $$\frac{1}{18}$$

55. Calculate this operation before combining the remaining terms.

    $$1\times \frac{1}{18}=\frac{1}{18}$$

56. For multiplication of fractions, multiply the numerators together and the denominators together.

    $$\frac{1\times \left(-3\right)}{18\times 1}$$

57. Calculate these two products separately.

    $$-\frac{3}{18}$$

58. Show the two divisions explicitly.

    $$\frac{-\frac{3}{3}}{\frac{18}{3}}$$

59. Evaluate the denominator division.

    $$\frac{18}{3}=6$$

60. Write the reduced fraction.

    $$-\frac{1}{6}$$

61. To add these fractions, first use a common denominator: the least common multiple of their denominators.

    $$\mathrm{LCD}=6$$

62. Multiply this numerator AND denominator by the same factor. Its value stays unchanged.

    $$\frac{5}{3}=\frac{5\times 2}{3\times 2}$$

63. Calculate the converted numerator; the denominator is now common.

    $$\frac{10}{6}$$

64. Multiply this numerator AND denominator by the same factor. Its value stays unchanged.

    $$-\frac{1}{6}=\frac{\left(-1\right)\times 1}{6\times 1}$$

65. Keep the common denominator and add the signed numerators.

    $$\frac{10-1}{6}=\frac{9}{6}$$

66. Show the two divisions explicitly.

    $$\frac{\frac{9}{3}}{\frac{6}{3}}$$

67. Evaluate the numerator division.

    $$\frac{9}{3}=3$$

68. Evaluate the denominator division.

    $$\frac{6}{3}=2$$

69. Write the reduced fraction.

    $$\frac{3}{2}$$

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

    $$-(-\frac{5}{3})+\frac{1}{18}\times \left(-3\right)$$

71. For multiplication of fractions, multiply the numerators together and the denominators together.

    $$\frac{\left(-1\right)\times 1}{1\times 9}$$

72. Calculate this operation before combining the remaining terms.

    $$\left(-1\right)\times \frac{1}{9}=-\frac{1}{9}$$

73. For multiplication of fractions, multiply the numerators together and the denominators together.

    $$\frac{\left(-1\right)\times \left(-15\right)}{9\times 1}$$

74. Calculate these two products separately.

    $$\frac{15}{9}$$

75. Show the two divisions explicitly.

    $$\frac{\frac{15}{3}}{\frac{9}{3}}$$

76. Evaluate the numerator division.

    $$\frac{15}{3}=5$$

77. Calculate this individual product, including its signs.

    $$\left(-\frac{1}{9}\right)\times \left(-15\right)=\frac{5}{3}$$

78. Calculate this individual product, including its signs.

    $$\frac{1}{18}\times \left(-3\right)=-\frac{1}{6}$$

79. For multiplication of fractions, multiply the numerators together and the denominators together.

    $$\frac{1\times 1}{1\times 6}$$

80. Calculate these two products separately.

    $$\frac{1}{6}$$

81. Calculate this operation before combining the remaining terms.

    $$1\times \frac{1}{6}=\frac{1}{6}$$

82. For multiplication of fractions, multiply the numerators together and the denominators together.

    $$\frac{\left(-1\right)\times 1}{1\times 6}$$

83. Calculate this operation before combining the remaining terms.

    $$\left(-1\right)\times \frac{1}{6}=-\frac{1}{6}$$

84. Add the calculated products to obtain this entry.

    $$\frac{5}{3}-\frac{1}{6}=\frac{3}{2}$$

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

    $$\left[\begin{array}{c}-6\\ \frac{3}{2}\end{array}\right]$$

86. Calculate this operation before combining the remaining terms.

    $$1\times \left(-6\right)=-6$$

87. Calculate this operation before combining the remaining terms.

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

88. Calculate this operation before combining the remaining terms.

    $$-6-9=-15$$

89. Check in this original equation by multiplying the coefficients by the calculated values.

    $$1\times \left(-6\right)-9=-15$$

90. Calculate this operation before combining the remaining terms.

    $$2\times \left(-6\right)=-12$$

91. For multiplication of fractions, multiply the numerators together and the denominators together.

    $$\frac{3\times 1}{1\times 2}$$

92. Calculate this operation before combining the remaining terms.

    $$3\times \frac{1}{2}=\frac{3}{2}$$

93. For multiplication of fractions, multiply the numerators together and the denominators together.

    $$\frac{6\times 3}{1\times 2}$$

94. Calculate these two products separately.

    $$\frac{18}{2}$$

95. Show the two divisions explicitly.

    $$\frac{\frac{18}{2}}{\frac{2}{2}}$$

96. Evaluate the denominator division.

    $$\frac{2}{2}=1$$

97. Write the reduced fraction.

    $$9\times \frac{1}{1}$$

98. Calculate this operation before combining the remaining terms.

    $$6\times \frac{3}{2}=9$$

99. Calculate this operation before combining the remaining terms.

    $$-12+9=-3$$

100. Check in this original equation by multiplying the coefficients by the calculated values.

     $$2\times \left(-6\right)+6\times \frac{3}{2}=-3$$

Answer

The final answer is shown below.