Mathematics 10 — Exercise 3.5 · Q5 (iv)

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.

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

2. Calculate the first diagonal product.

   $$3\times \left(-3\right)=-9$$

3. Calculate the second diagonal product.

   $$5\times 5=25$$

4. Subtract the second product, including its sign.

   $$D=-9-25$$

5. Simplify the subtraction.

   $$D=-34$$

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

   $$-34\ne 0$$

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

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

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

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

9. Calculate this operation before combining the remaining terms.

   $$\frac{1}{-34}=-\frac{1}{34}$$

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

    $$\frac{\left(-3\right)\times \left(-1\right)}{1\times 34}$$

11. Calculate these two products separately.

    $$\frac{3}{34}$$

12. Calculate this operation before combining the remaining terms.

    $$\left(-3\right)\times \left(-\frac{1}{34}\right)=\frac{3}{34}$$

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

    $$-\frac{3}{-34}=\frac{3}{34}$$

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

    $$\frac{\left(-5\right)\times \left(-1\right)}{1\times 34}$$

15. Calculate these two products separately.

    $$\frac{5}{34}$$

16. Calculate this operation before combining the remaining terms.

    $$\left(-5\right)\times \left(-\frac{1}{34}\right)=\frac{5}{34}$$

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

    $$-\frac{5}{-34}=\frac{5}{34}$$

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

    $$-\frac{5}{-34}=\frac{5}{34}$$

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

    $$\frac{3\times \left(-1\right)}{1\times 34}$$

20. Calculate these two products separately.

    $$-\frac{3}{34}$$

21. Calculate this operation before combining the remaining terms.

    $$3\times \left(-\frac{1}{34}\right)=-\frac{3}{34}$$

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

    $$\frac{3}{-34}=-\frac{3}{34}$$

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

    $$\left[\begin{array}{cc}\frac{3}{34}& \frac{5}{34}\\ \frac{5}{34}& -\frac{3}{34}\end{array}\right]$$

Answer

The final answer is shown below.