Mathematics 10 — Exercise Review · Q5 (i) 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.

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

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

   $$D=3\times \left(-1\right)-4\times 5$$

3. Calculate the first diagonal product.

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

4. Calculate the second diagonal product.

   $$4\times 5=20$$

5. Subtract the second product, including its sign.

   $$D=-3-20$$

6. Simplify the subtraction.

   $$D=-23$$

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

   $$D=3\times \left(-1\right)-4\times 5$$

8. Calculate the first diagonal product.

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

9. Calculate the second diagonal product.

   $$4\times 5=20$$

10. Subtract the second product, including its sign.

    $$D=-3-20$$

11. Simplify the subtraction.

    $$D=-23$$

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

    $$-23\ne 0$$

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

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

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

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

15. Calculate this operation before combining the remaining terms.

    $$\frac{1}{-23}=-\frac{1}{23}$$

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

    $$\frac{\left(-1\right)\times \left(-1\right)}{1\times 23}$$

17. Calculate these two products separately.

    $$\frac{1}{23}$$

18. Calculate this operation before combining the remaining terms.

    $$\left(-1\right)\times \left(-\frac{1}{23}\right)=\frac{1}{23}$$

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

    $$-\frac{1}{-23}=\frac{1}{23}$$

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

    $$\frac{\left(-4\right)\times \left(-1\right)}{1\times 23}$$

21. Calculate these two products separately.

    $$\frac{4}{23}$$

22. Calculate this operation before combining the remaining terms.

    $$\left(-4\right)\times \left(-\frac{1}{23}\right)=\frac{4}{23}$$

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

    $$-\frac{4}{-23}=\frac{4}{23}$$

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

    $$\frac{\left(-5\right)\times \left(-1\right)}{1\times 23}$$

25. Calculate these two products separately.

    $$\frac{5}{23}$$

26. Calculate this operation before combining the remaining terms.

    $$\left(-5\right)\times \left(-\frac{1}{23}\right)=\frac{5}{23}$$

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

    $$-\frac{5}{-23}=\frac{5}{23}$$

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

    $$\frac{3\times \left(-1\right)}{1\times 23}$$

29. Calculate these two products separately.

    $$-\frac{3}{23}$$

30. Calculate this operation before combining the remaining terms.

    $$3\times \left(-\frac{1}{23}\right)=-\frac{3}{23}$$

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

    $$\frac{3}{-23}=-\frac{3}{23}$$

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

    $$\left[\begin{array}{cc}\frac{1}{23}& \frac{4}{23}\\ \frac{5}{23}& -\frac{3}{23}\end{array}\right]$$

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

34. 2 columns in the first matrix match 2 rows in the second. The product has 2 rows and 1 columns.

    $$(\left[\begin{array}{cc}\frac{1}{23}& \frac{4}{23}\\ \frac{5}{23}& -\frac{3}{23}\end{array}\right],\left[\begin{array}{c}7\\ 2\end{array}\right])$$

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

    $$\frac{1\times 1}{1\times 23}$$

36. Calculate this operation before combining the remaining terms.

    $$1\times \frac{1}{23}=\frac{1}{23}$$

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

    $$\frac{1\times 7}{23\times 1}$$

38. Calculate these two products separately.

    $$\frac{7}{23}$$

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

    $$\frac{4\times 1}{1\times 23}$$

40. Calculate this operation before combining the remaining terms.

    $$4\times \frac{1}{23}=\frac{4}{23}$$

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

    $$\frac{4\times 2}{23\times 1}$$

42. Calculate these two products separately.

    $$\frac{8}{23}$$

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

    $$\mathrm{LCD}=23$$

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

    $$\frac{7}{23}=\frac{7\times 1}{23\times 1}$$

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

    $$\frac{8}{23}=\frac{8\times 1}{23\times 1}$$

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

    $$\frac{7+8}{23}=\frac{15}{23}$$

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

    $$\frac{1}{23}\times 7+\frac{4}{23}\times 2$$

48. Calculate this individual product, including its signs.

    $$\frac{1}{23}\times 7=\frac{7}{23}$$

49. Calculate this individual product, including its signs.

    $$\frac{4}{23}\times 2=\frac{8}{23}$$

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

    $$\frac{7\times 1}{1\times 23}$$

51. Calculate this operation before combining the remaining terms.

    $$7\times \frac{1}{23}=\frac{7}{23}$$

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

    $$\frac{8\times 1}{1\times 23}$$

53. Calculate this operation before combining the remaining terms.

    $$8\times \frac{1}{23}=\frac{8}{23}$$

54. Add the calculated products to obtain this entry.

    $$\frac{7}{23}+\frac{8}{23}=\frac{15}{23}$$

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

    $$\frac{5\times 1}{1\times 23}$$

56. Calculate this operation before combining the remaining terms.

    $$5\times \frac{1}{23}=\frac{5}{23}$$

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

    $$\frac{5\times 7}{23\times 1}$$

58. Calculate these two products separately.

    $$\frac{35}{23}$$

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

    $$\frac{6\times 1}{1\times 23}$$

60. Calculate these two products separately.

    $$\frac{6}{23}$$

61. Calculate this operation before combining the remaining terms.

    $$6\times \frac{1}{23}=\frac{6}{23}$$

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

    $$\frac{\left(-1\right)\times 6}{1\times 23}$$

63. Calculate these two products separately.

    $$-\frac{6}{23}$$

64. Calculate this operation before combining the remaining terms.

    $$\left(-1\right)\times \frac{6}{23}=-\frac{6}{23}$$

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

    $$\frac{35}{23}=\frac{35\times 1}{23\times 1}$$

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

    $$-\frac{6}{23}=\frac{\left(-6\right)\times 1}{23\times 1}$$

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

    $$\frac{35-6}{23}=\frac{29}{23}$$

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

    $$\frac{5}{23}\times 7-\frac{6}{23}$$

69. Calculate this individual product, including its signs.

    $$\frac{5}{23}\times 7=\frac{35}{23}$$

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

    $$\frac{\left(-3\right)\times 1}{1\times 23}$$

71. Calculate this operation before combining the remaining terms.

    $$\left(-3\right)\times \frac{1}{23}=-\frac{3}{23}$$

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

    $$\frac{\left(-3\right)\times 2}{23\times 1}$$

73. Calculate this individual product, including its signs.

    $$\left(-\frac{3}{23}\right)\times 2=-\frac{6}{23}$$

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

    $$\frac{35\times 1}{1\times 23}$$

75. Calculate this operation before combining the remaining terms.

    $$35\times \frac{1}{23}=\frac{35}{23}$$

76. Add the calculated products to obtain this entry.

    $$\frac{35}{23}-\frac{6}{23}=\frac{29}{23}$$

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

    $$\left[\begin{array}{c}\frac{15}{23}\\ \frac{29}{23}\end{array}\right]$$

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

    $$\frac{15\times 1}{1\times 23}$$

79. Calculate these two products separately.

    $$\frac{15}{23}$$

80. Calculate this operation before combining the remaining terms.

    $$15\times \frac{1}{23}=\frac{15}{23}$$

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

    $$\frac{3\times 15}{1\times 23}$$

82. Calculate these two products separately.

    $$\frac{45}{23}$$

83. Calculate this operation before combining the remaining terms.

    $$3\times \frac{15}{23}=\frac{45}{23}$$

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

    $$\frac{29\times 1}{1\times 23}$$

85. Calculate these two products separately.

    $$\frac{29}{23}$$

86. Calculate this operation before combining the remaining terms.

    $$29\times \frac{1}{23}=\frac{29}{23}$$

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

    $$\frac{4\times 29}{1\times 23}$$

88. Calculate these two products separately.

    $$\frac{116}{23}$$

89. Calculate this operation before combining the remaining terms.

    $$4\times \frac{29}{23}=\frac{116}{23}$$

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

    $$\frac{45}{23}=\frac{45\times 1}{23\times 1}$$

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

    $$\frac{116}{23}=\frac{116\times 1}{23\times 1}$$

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

    $$\frac{45+116}{23}=\frac{161}{23}$$

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

    $$\mathrm{GCD}=23$$

94. Show the two divisions explicitly.

    $$\frac{\frac{161}{23}}{\frac{23}{23}}$$

95. Evaluate the numerator division.

    $$\frac{161}{23}=7$$

96. Evaluate the denominator division.

    $$\frac{23}{23}=1$$

97. Write the reduced fraction.

    $$7\times \frac{1}{1}$$

98. Calculate this operation before combining the remaining terms.

    $$\frac{45}{23}+\frac{116}{23}=7$$

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

    $$3\times \frac{15}{23}+4\times \frac{29}{23}=7$$

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

     $$\frac{5\times 15}{1\times 23}$$

101. Calculate these two products separately.

     $$\frac{75}{23}$$

102. Calculate this operation before combining the remaining terms.

     $$5\times \frac{15}{23}=\frac{75}{23}$$

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

     $$\frac{\left(-1\right)\times 29}{1\times 23}$$

104. Calculate these two products separately.

     $$-\frac{29}{23}$$

105. Calculate this operation before combining the remaining terms.

     $$\left(-1\right)\times \frac{29}{23}=-\frac{29}{23}$$

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

     $$\frac{75}{23}=\frac{75\times 1}{23\times 1}$$

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

     $$-\frac{29}{23}=\frac{\left(-29\right)\times 1}{23\times 1}$$

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

     $$\frac{75-29}{23}=\frac{46}{23}$$

109. Show the two divisions explicitly.

     $$\frac{\frac{46}{23}}{\frac{23}{23}}$$

110. Evaluate the numerator division.

     $$\frac{46}{23}=2$$

111. Write the reduced fraction.

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

112. Calculate this operation before combining the remaining terms.

     $$\frac{75}{23}-\frac{29}{23}=2$$

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

     $$5\times \frac{15}{23}-\frac{29}{23}=2$$

Answer

The final answer is shown below.