Mathematics 10 — Exercise 3.3 · Q6 (i)

Full question

Verify (A+B)ᵗ = Aᵗ+Bᵗ.

Step-by-step solution

1. Compute the left side first, following the bracket order.

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

   $$(\left[\begin{array}{cc}6& -2\\ 0& 3\end{array}\right],\left[\begin{array}{cc}8& -1\\ 3& 0\end{array}\right])$$

3. Entry (1,1): add the corresponding entries.

   $$6+8=14$$

4. Calculate this operation before combining the remaining terms.

   $$\left(-1\right)\times 1=-1$$

5. Entry (1,2): add the corresponding entries.

   $$-2-1=-3$$

6. Entry (2,1): add the corresponding entries.

   $$0+3=3$$

7. Entry (2,2): add the corresponding entries.

   $$3+0=3$$

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

   $$\left[\begin{array}{cc}14& -3\\ 3& 3\end{array}\right]$$

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

   $$\left[\begin{array}{cc}14& -3\\ 3& 3\end{array}\right]$$

10. Move row 1 into column 1.

    $$\left[\begin{array}{c}14\\ -3\end{array}\right]$$

11. Move row 2 into column 2.

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

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

    $$\left[\begin{array}{cc}14& 3\\ -3& 3\end{array}\right]$$

13. Now compute the right side independently.

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

    $$\left[\begin{array}{cc}6& -2\\ 0& 3\end{array}\right]$$

15. Move row 1 into column 1.

    $$\left[\begin{array}{c}6\\ -2\end{array}\right]$$

16. Move row 2 into column 2.

    $$\left[\begin{array}{c}0\\ 3\end{array}\right]$$

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

    $$\left[\begin{array}{cc}6& 0\\ -2& 3\end{array}\right]$$

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

    $$\left[\begin{array}{cc}8& -1\\ 3& 0\end{array}\right]$$

19. Move row 1 into column 1.

    $$\left[\begin{array}{c}8\\ -1\end{array}\right]$$

20. Move row 2 into column 2.

    $$\left[\begin{array}{c}3\\ 0\end{array}\right]$$

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

    $$\left[\begin{array}{cc}8& 3\\ -1& 0\end{array}\right]$$

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

    $$(\left[\begin{array}{cc}6& 0\\ -2& 3\end{array}\right],\left[\begin{array}{cc}8& 3\\ -1& 0\end{array}\right])$$

23. Entry (1,1): add the corresponding entries.

    $$6+8=14$$

24. Entry (1,2): add the corresponding entries.

    $$0+3=3$$

25. Entry (2,1): add the corresponding entries.

    $$-2-1=-3$$

26. Entry (2,2): add the corresponding entries.

    $$3+0=3$$

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

    $$\left[\begin{array}{cc}14& 3\\ -3& 3\end{array}\right]$$

28. Every corresponding entry is equal, so the stated identity holds.

    $$(\left[\begin{array}{cc}14& 3\\ -3& 3\end{array}\right],\left[\begin{array}{cc}14& 3\\ -3& 3\end{array}\right])$$

Answer

Every corresponding entry is equal, so the stated identity holds.