Mathematics 10 — Exercise 3.5 · Q5 (i)

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.

   $$D=5\times 5-0\times 0$$

2. Calculate the first diagonal product.

   $$5\times 5=25$$

3. Calculate the second diagonal product.

   $$0\times 0=0$$

4. Subtract the second product, including its sign.

   $$D=25+0$$

5. Simplify the subtraction.

   $$D=25$$

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

   $$25\ne 0$$

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

   $$\left[\begin{array}{cc}5& 0\\ 0& 5\end{array}\right]$$

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

   $$\left[\begin{array}{cc}5& 0\\ 0& 5\end{array}\right]$$

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

   $$\frac{5}{25}=\frac{1}{5}$$

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

    $$\frac{0}{25}=0$$

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

    $$\frac{0}{25}=0$$

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

    $$\frac{5}{25}=\frac{1}{5}$$

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

    $$\left[\begin{array}{cc}\frac{1}{5}& 0\\ 0& \frac{1}{5}\end{array}\right]$$

Answer

The final answer is shown below.