Mathematics 10 — Exercise 3.5 · Q6

Full question

Find A⁻¹ and verify AA⁻¹ = A⁻¹A = I.

Step-by-step solution

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

   $$D=5\times \left(-1\right)-\left(-3\right)\times 2$$

2. Calculate the first diagonal product.

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

3. Calculate the second diagonal product.

   $$\left(-3\right)\times 2=-6$$

4. Subtract the second product, including its sign.

   $$D=-5+6$$

5. Simplify the subtraction.

   $$D=1$$

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

   $$1\ne 0$$

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

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

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

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

9. Calculate this operation before combining the remaining terms.

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

10. Calculate this operation before combining the remaining terms.

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

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

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

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

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

13. Calculate this operation before combining the remaining terms.

    $$1\times 1=1$$

14. Calculate this operation before combining the remaining terms.

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

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

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

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

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

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

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

18. 2 columns in the first matrix match 2 rows in the second. The product has 2 rows and 2 columns.

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

19. Calculate this operation before combining the remaining terms.

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

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

    $$5\times \left(-1\right)--6$$

21. Calculate this individual product, including its signs.

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

22. Calculate this individual product, including its signs.

    $$\left(-3\right)\times \left(-2\right)=6$$

23. Add the calculated products to obtain this entry.

    $$-5+6=1$$

24. Calculate this operation before combining the remaining terms.

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

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

    $$5\times 3-15$$

26. Calculate this individual product, including its signs.

    $$5\times 3=15$$

27. Calculate this individual product, including its signs.

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

28. Add the calculated products to obtain this entry.

    $$15-15=0$$

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

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

30. Calculate this individual product, including its signs.

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

31. Calculate this individual product, including its signs.

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

32. Add the calculated products to obtain this entry.

    $$-2+2=0$$

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

    $$2\times 3-5$$

34. Calculate this individual product, including its signs.

    $$2\times 3=6$$

35. Calculate this individual product, including its signs.

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

36. Add the calculated products to obtain this entry.

    $$6-5=1$$

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

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

38. 2 columns in the first matrix match 2 rows in the second. The product has 2 rows and 2 columns.

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

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

    $$-5+3\times 2$$

40. Calculate this individual product, including its signs.

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

41. Calculate this individual product, including its signs.

    $$3\times 2=6$$

42. Add the calculated products to obtain this entry.

    $$-5+6=1$$

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

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

44. Calculate this individual product, including its signs.

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

45. Calculate this individual product, including its signs.

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

46. Add the calculated products to obtain this entry.

    $$3-3=0$$

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

    $$-10+5\times 2$$

48. Calculate this individual product, including its signs.

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

49. Calculate this individual product, including its signs.

    $$5\times 2=10$$

50. Add the calculated products to obtain this entry.

    $$-10+10=0$$

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

    $$--6+5\times \left(-1\right)$$

52. Calculate this individual product, including its signs.

    $$\left(-2\right)\times \left(-3\right)=6$$

53. Calculate this individual product, including its signs.

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

54. Add the calculated products to obtain this entry.

    $$6-5=1$$

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

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

Answer

The final answer is shown below.