Mathematics 10 — Exercise 3.5 · Q5 (vi)

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.

   $$D=-10-\left(-3\right)\times 4$$

2. Calculate the first diagonal product.

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

3. Calculate the second diagonal product.

   $$\left(-3\right)\times 4=-12$$

4. Subtract the second product, including its sign.

   $$D=-10+12$$

5. Simplify the subtraction.

   $$D=2$$

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

   $$2\ne 0$$

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

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

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

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

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

   $$\frac{5}{2}=\frac{5}{2}$$

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

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

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

    $$\frac{\left(-4\right)\times 1}{1\times 2}$$

12. Calculate these two products separately.

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

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

    $$\mathrm{GCD}=2$$

14. Show the two divisions explicitly.

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

15. Evaluate the denominator division.

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

16. Write the reduced fraction.

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

17. Calculate this operation before combining the remaining terms.

    $$\left(-4\right)\times \frac{1}{2}=-2$$

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

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

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

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

20. Calculate these two products separately.

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

21. Show the two divisions explicitly.

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

22. Write the reduced fraction.

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

23. Calculate this operation before combining the remaining terms.

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

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

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

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

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

Answer

The final answer is shown below.