Mathematics 10 — Exercise Review · Q4 (iv)

Full question

Find A⁻¹A.

Step-by-step solution

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

   $$D=7\times 1-3\times 2$$

2. Calculate the first diagonal product.

   $$7\times 1=7$$

3. Calculate the second diagonal product.

   $$3\times 2=6$$

4. Subtract the second product, including its sign.

   $$D=7-6$$

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.

   $$\left[\begin{array}{cc}1& 3\\ 2& 7\end{array}\right]$$

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

   $$\left[\begin{array}{cc}1& -3\\ -2& 7\end{array}\right]$$

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

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

10. Calculate this operation before combining the remaining terms.

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

11. Calculate this operation before combining the remaining terms.

    $$1\times 1=1$$

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

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

13. Calculate this operation before combining the remaining terms.

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

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

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

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

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

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

    $$\left[\begin{array}{cc}1& -3\\ -2& 7\end{array}\right]$$

17. 2 columns in the first matrix match 2 rows in the second. The product has 2 rows and 2 columns.

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

18. Calculate this operation before combining the remaining terms.

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

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

    $$1\times 7-6$$

20. Calculate this individual product, including its signs.

    $$1\times 7=7$$

21. Calculate this individual product, including its signs.

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

22. Add the calculated products to obtain this entry.

    $$7-6=1$$

23. Calculate this operation before combining the remaining terms.

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

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

    $$1\times 3-3$$

25. Calculate this individual product, including its signs.

    $$1\times 3=3$$

26. Calculate this individual product, including its signs.

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

27. Add the calculated products to obtain this entry.

    $$3-3=0$$

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

    $$-14+7\times 2$$

29. Calculate this individual product, including its signs.

    $$\left(-2\right)\times 7=-14$$

30. Calculate this individual product, including its signs.

    $$7\times 2=14$$

31. Add the calculated products to obtain this entry.

    $$-14+14=0$$

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

    $$-6+7\times 1$$

33. Calculate this individual product, including its signs.

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

34. Calculate this individual product, including its signs.

    $$7\times 1=7$$

35. Add the calculated products to obtain this entry.

    $$-6+7=1$$

36. 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.