Mathematics 10 — Exercise Review · Q2 (i)

Full question

Find (A−B)ᵗ.

Step-by-step solution

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

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

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

   $$4-5=-1$$

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

   $$2-5=-3$$

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

   $$7-8=-1$$

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

   $$6-8=-2$$

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

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

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

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

8. Move row 1 into column 1.

   $$\left[\begin{array}{c}-1\\ -3\end{array}\right]$$

9. Move row 2 into column 2.

   $$\left[\begin{array}{c}-1\\ -2\end{array}\right]$$

10. 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.