Mathematics 10 — Exercise Review · Q2 (ii)

Full question

Find Bᵗ−Aᵗ.

Step-by-step solution

1. Transpose: row 1 becomes column 1, row 2 becomes column 2, and so on.

   $$\left[\begin{array}{cc}5& 5\\ 8& 8\end{array}\right]$$

2. Move row 1 into column 1.

   $$\left[\begin{array}{c}5\\ 5\end{array}\right]$$

3. Move row 2 into column 2.

   $$\left[\begin{array}{c}8\\ 8\end{array}\right]$$

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

   $$\left[\begin{array}{cc}5& 8\\ 5& 8\end{array}\right]$$

5. Transpose: row 1 becomes column 1, row 2 becomes column 2, and so on.

   $$\left[\begin{array}{cc}4& 2\\ 7& 6\end{array}\right]$$

6. Move row 1 into column 1.

   $$\left[\begin{array}{c}4\\ 2\end{array}\right]$$

7. Move row 2 into column 2.

   $$\left[\begin{array}{c}7\\ 6\end{array}\right]$$

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

   $$\left[\begin{array}{cc}4& 7\\ 2& 6\end{array}\right]$$

9. The orders match. Combine only entries in the same row and column.

   $$(\left[\begin{array}{cc}5& 8\\ 5& 8\end{array}\right],\left[\begin{array}{cc}4& 7\\ 2& 6\end{array}\right])$$

10. Entry (1,1): subtract the corresponding entries.

    $$5-4=1$$

11. Entry (1,2): subtract the corresponding entries.

    $$8-7=1$$

12. Entry (2,1): subtract the corresponding entries.

    $$5-2=3$$

13. Entry (2,2): subtract the corresponding entries.

    $$8-6=2$$

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

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

Answer

The final answer is shown below.