Mathematics 10 — Exercise 3.5 · Q5 (ii)

Full question

Find the multiplicative inverse of the following matrix.

Step-by-step solution

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

   $$D=-8-8\times 7$$

2. Calculate the first diagonal product.

   $$\left(-4\right)\times 2=-8$$

3. Calculate the second diagonal product.

   $$8\times 7=56$$

4. Subtract the second product, including its sign.

   $$D=-8-56$$

5. Simplify the subtraction.

   $$D=-64$$

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

   $$-64\ne 0$$

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

   $$\left[\begin{array}{cc}2& 8\\ 7& -4\end{array}\right]$$

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

   $$\left[\begin{array}{cc}2& -8\\ -7& -4\end{array}\right]$$

9. Calculate this operation before combining the remaining terms.

   $$\frac{1}{-64}=-\frac{1}{64}$$

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

    $$\frac{2\times \left(-1\right)}{1\times 64}$$

11. Calculate these two products separately.

    $$-\frac{2}{64}$$

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

    $$\mathrm{GCD}=2$$

13. Show the two divisions explicitly.

    $$\frac{-\frac{2}{2}}{\frac{64}{2}}$$

14. Evaluate the numerator division.

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

15. Evaluate the denominator division.

    $$\frac{64}{2}=32$$

16. Write the reduced fraction.

    $$-\frac{1}{32}$$

17. Calculate this operation before combining the remaining terms.

    $$2\times \left(-\frac{1}{64}\right)=-\frac{1}{32}$$

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

    $$\frac{2}{-64}=-\frac{1}{32}$$

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

    $$\frac{\left(-8\right)\times \left(-1\right)}{1\times 64}$$

20. Calculate these two products separately.

    $$\frac{8}{64}$$

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

    $$\mathrm{GCD}=8$$

22. Show the two divisions explicitly.

    $$\frac{\frac{8}{8}}{\frac{64}{8}}$$

23. Evaluate the numerator division.

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

24. Evaluate the denominator division.

    $$\frac{64}{8}=8$$

25. Write the reduced fraction.

    $$\frac{1}{8}$$

26. Calculate this operation before combining the remaining terms.

    $$\left(-8\right)\times \left(-\frac{1}{64}\right)=\frac{1}{8}$$

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

    $$-\frac{8}{-64}=\frac{1}{8}$$

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

    $$\frac{\left(-7\right)\times \left(-1\right)}{1\times 64}$$

29. Calculate these two products separately.

    $$\frac{7}{64}$$

30. Calculate this operation before combining the remaining terms.

    $$\left(-7\right)\times \left(-\frac{1}{64}\right)=\frac{7}{64}$$

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

    $$-\frac{7}{-64}=\frac{7}{64}$$

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

    $$\frac{\left(-4\right)\times \left(-1\right)}{1\times 64}$$

33. Calculate these two products separately.

    $$\frac{4}{64}$$

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

    $$\mathrm{GCD}=4$$

35. Show the two divisions explicitly.

    $$\frac{\frac{4}{4}}{\frac{64}{4}}$$

36. Evaluate the numerator division.

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

37. Evaluate the denominator division.

    $$\frac{64}{4}=16$$

38. Write the reduced fraction.

    $$\frac{1}{16}$$

39. Calculate this operation before combining the remaining terms.

    $$\left(-4\right)\times \left(-\frac{1}{64}\right)=\frac{1}{16}$$

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

    $$-\frac{4}{-64}=\frac{1}{16}$$

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

    $$\left[\begin{array}{cc}-\frac{1}{32}& \frac{1}{8}\\ \frac{7}{64}& \frac{1}{16}\end{array}\right]$$

Answer

The final answer is shown below.