Mathematics 10 — Exercise Review · Q3 (i)

Full question

Verify 2(A+B)=2A+2B.

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.

   $$(\left[\begin{array}{cc}2& 3\\ 4& 5\end{array}\right],\left[\begin{array}{cc}5& 6\\ 2& 1\end{array}\right])$$

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

   $$2+5=7$$

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

   $$3+6=9$$

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

   $$4+2=6$$

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

   $$5+1=6$$

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

   $$\left[\begin{array}{cc}7& 9\\ 6& 6\end{array}\right]$$

8. Multiply every matrix entry by the scalar.

   $$(2,\left[\begin{array}{cc}7& 9\\ 6& 6\end{array}\right])$$

9. Entry (1,1): multiply its value by the scalar.

   $$2\times 7=14$$

10. Entry (1,2): multiply its value by the scalar.

    $$2\times 9=18$$

11. Entry (2,1): multiply its value by the scalar.

    $$2\times 6=12$$

12. Entry (2,2): multiply its value by the scalar.

    $$2\times 6=12$$

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

    $$\left[\begin{array}{cc}14& 18\\ 12& 12\end{array}\right]$$

14. Now compute the right side independently.

15. Multiply every matrix entry by the scalar.

    $$(2,\left[\begin{array}{cc}2& 3\\ 4& 5\end{array}\right])$$

16. Entry (1,1): multiply its value by the scalar.

    $$2\times 2=4$$

17. Entry (1,2): multiply its value by the scalar.

    $$2\times 3=6$$

18. Entry (2,1): multiply its value by the scalar.

    $$2\times 4=8$$

19. Entry (2,2): multiply its value by the scalar.

    $$2\times 5=10$$

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

    $$\left[\begin{array}{cc}4& 6\\ 8& 10\end{array}\right]$$

21. Multiply every matrix entry by the scalar.

    $$(2,\left[\begin{array}{cc}5& 6\\ 2& 1\end{array}\right])$$

22. Entry (1,1): multiply its value by the scalar.

    $$2\times 5=10$$

23. Entry (1,2): multiply its value by the scalar.

    $$2\times 6=12$$

24. Entry (2,1): multiply its value by the scalar.

    $$2\times 2=4$$

25. Entry (2,2): multiply its value by the scalar.

    $$2\times 1=2$$

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

    $$\left[\begin{array}{cc}10& 12\\ 4& 2\end{array}\right]$$

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

    $$(\left[\begin{array}{cc}4& 6\\ 8& 10\end{array}\right],\left[\begin{array}{cc}10& 12\\ 4& 2\end{array}\right])$$

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

    $$4+10=14$$

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

    $$6+12=18$$

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

    $$8+4=12$$

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

    $$10+2=12$$

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

    $$\left[\begin{array}{cc}14& 18\\ 12& 12\end{array}\right]$$

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

    $$(\left[\begin{array}{cc}14& 18\\ 12& 12\end{array}\right],\left[\begin{array}{cc}14& 18\\ 12& 12\end{array}\right])$$

Answer

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