Mathematics 10 — Exercise 3.4 · Q2 (i)

Full question

Verify AB ≠ BA.

Step-by-step solution

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

2. 2 columns in the first matrix match 2 rows in the second. The product has 2 rows and 2 columns.

   $$(\left[\begin{array}{cc}5& 1\\ -1& 4\end{array}\right],\left[\begin{array}{cc}2& -1\\ 0& 3\end{array}\right])$$

3. Entry (1,1): pair row 1 with column 1 and add their products.

   $$5\times 2+1\times 0$$

4. Calculate this individual product, including its signs.

   $$5\times 2=10$$

5. Calculate this individual product, including its signs.

   $$1\times 0=0$$

6. Add the calculated products to obtain this entry.

   $$10+0=10$$

7. Entry (1,2): pair row 1 with column 2 and add their products.

   $$5\times \left(-1\right)+1\times 3$$

8. Calculate this individual product, including its signs.

   $$5\times \left(-1\right)=-5$$

9. Calculate this individual product, including its signs.

   $$1\times 3=3$$

10. Add the calculated products to obtain this entry.

    $$-5+3=-2$$

11. Entry (2,1): pair row 2 with column 1 and add their products.

    $$-2+4\times 0$$

12. Calculate this individual product, including its signs.

    $$\left(-1\right)\times 2=-2$$

13. Calculate this individual product, including its signs.

    $$4\times 0=0$$

14. Add the calculated products to obtain this entry.

    $$-2+0=-2$$

15. Entry (2,2): pair row 2 with column 2 and add their products.

    $$--1+4\times 3$$

16. Calculate this individual product, including its signs.

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

17. Calculate this individual product, including its signs.

    $$4\times 3=12$$

18. Add the calculated products to obtain this entry.

    $$1+12=13$$

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

    $$\left[\begin{array}{cc}10& -2\\ -2& 13\end{array}\right]$$

20. Now compute the right side independently.

21. 2 columns in the first matrix match 2 rows in the second. The product has 2 rows and 2 columns.

    $$(\left[\begin{array}{cc}2& -1\\ 0& 3\end{array}\right],\left[\begin{array}{cc}5& 1\\ -1& 4\end{array}\right])$$

22. Entry (1,1): pair row 1 with column 1 and add their products.

    $$2\times 5--1$$

23. Calculate this individual product, including its signs.

    $$2\times 5=10$$

24. Calculate this individual product, including its signs.

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

25. Add the calculated products to obtain this entry.

    $$10+1=11$$

26. Entry (1,2): pair row 1 with column 2 and add their products.

    $$2\times 1-4$$

27. Calculate this individual product, including its signs.

    $$2\times 1=2$$

28. Calculate this individual product, including its signs.

    $$\left(-1\right)\times 4=-4$$

29. Add the calculated products to obtain this entry.

    $$2-4=-2$$

30. Entry (2,1): pair row 2 with column 1 and add their products.

    $$0\times 5+3\times \left(-1\right)$$

31. Calculate this individual product, including its signs.

    $$0\times 5=0$$

32. Calculate this individual product, including its signs.

    $$3\times \left(-1\right)=-3$$

33. Add the calculated products to obtain this entry.

    $$0-3=-3$$

34. Entry (2,2): pair row 2 with column 2 and add their products.

    $$0\times 1+3\times 4$$

35. Calculate this individual product, including its signs.

    $$0\times 1=0$$

36. Calculate this individual product, including its signs.

    $$3\times 4=12$$

37. Add the calculated products to obtain this entry.

    $$0+12=12$$

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

    $$\left[\begin{array}{cc}11& -2\\ -3& 12\end{array}\right]$$

39. The two calculated matrices differ, as required.

    $$(\left[\begin{array}{cc}10& -2\\ -2& 13\end{array}\right],\left[\begin{array}{cc}11& -2\\ -3& 12\end{array}\right])$$

Answer

The two calculated matrices differ, as required.