Mathematics 10 — Exercise 3.5 · Q5 (v)

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.

   $$D=10\times 3-8\times 3$$

2. Calculate the first diagonal product.

   $$10\times 3=30$$

3. Calculate the second diagonal product.

   $$8\times 3=24$$

4. Subtract the second product, including its sign.

   $$D=30-24$$

5. Simplify the subtraction.

   $$D=6$$

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

   $$6\ne 0$$

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

   $$\left[\begin{array}{cc}3& 8\\ 3& 10\end{array}\right]$$

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

   $$\left[\begin{array}{cc}3& -8\\ -3& 10\end{array}\right]$$

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

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

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

    $$\frac{\left(-8\right)\times 1}{1\times 6}$$

11. Calculate these two products separately.

    $$-\frac{8}{6}$$

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

    $$\mathrm{GCD}=2$$

13. Show the two divisions explicitly.

    $$\frac{-\frac{8}{2}}{\frac{6}{2}}$$

14. Evaluate the numerator division.

    $$-\frac{8}{2}=-4$$

15. Evaluate the denominator division.

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

16. Write the reduced fraction.

    $$-\frac{4}{3}$$

17. Calculate this operation before combining the remaining terms.

    $$\left(-8\right)\times \frac{1}{6}=-\frac{4}{3}$$

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

    $$-\frac{8}{6}=-\frac{4}{3}$$

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

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

20. Calculate these two products separately.

    $$-\frac{3}{6}$$

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

    $$\mathrm{GCD}=3$$

22. Show the two divisions explicitly.

    $$\frac{-\frac{3}{3}}{\frac{6}{3}}$$

23. Evaluate the numerator division.

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

24. Evaluate the denominator division.

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

25. Write the reduced fraction.

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

26. Calculate this operation before combining the remaining terms.

    $$\left(-3\right)\times \frac{1}{6}=-\frac{1}{2}$$

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

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

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

    $$\frac{10}{6}=\frac{5}{3}$$

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

    $$\left[\begin{array}{cc}\frac{1}{2}& -\frac{4}{3}\\ -\frac{1}{2}& \frac{5}{3}\end{array}\right]$$

Answer

The final answer is shown below.