Mathematics 10 — Exercise 3.4 · Q2 (iv)

Full question

Verify (BC)ᵗ = CᵗBᵗ.

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.

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

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

   $$2\times 5-1$$

4. Calculate this individual product, including its signs.

   $$2\times 5=10$$

5. Calculate this individual product, including its signs.

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

6. Add the calculated products to obtain this entry.

   $$10-1=9$$

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

   $$2\times 1-4$$

8. Calculate this individual product, including its signs.

   $$2\times 1=2$$

9. Calculate this individual product, including its signs.

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

10. Add the calculated products to obtain this entry.

    $$2-4=-2$$

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

    $$0\times 5+3\times 1$$

12. Calculate this individual product, including its signs.

    $$0\times 5=0$$

13. Calculate this individual product, including its signs.

    $$3\times 1=3$$

14. Add the calculated products to obtain this entry.

    $$0+3=3$$

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

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

16. Calculate this individual product, including its signs.

    $$0\times 1=0$$

17. Calculate this individual product, including its signs.

    $$3\times 4=12$$

18. Add the calculated products to obtain this entry.

    $$0+12=12$$

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

    $$\left[\begin{array}{cc}9& -2\\ 3& 12\end{array}\right]$$

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

    $$\left[\begin{array}{cc}9& -2\\ 3& 12\end{array}\right]$$

21. Move row 1 into column 1.

    $$\left[\begin{array}{c}9\\ -2\end{array}\right]$$

22. Move row 2 into column 2.

    $$\left[\begin{array}{c}3\\ 12\end{array}\right]$$

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

    $$\left[\begin{array}{cc}9& 3\\ -2& 12\end{array}\right]$$

24. Now compute the right side independently.

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

    $$\left[\begin{array}{cc}5& 1\\ 1& 4\end{array}\right]$$

26. Move row 1 into column 1.

    $$\left[\begin{array}{c}5\\ 1\end{array}\right]$$

27. Move row 2 into column 2.

    $$\left[\begin{array}{c}1\\ 4\end{array}\right]$$

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

    $$\left[\begin{array}{cc}5& 1\\ 1& 4\end{array}\right]$$

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

    $$\left[\begin{array}{cc}2& -1\\ 0& 3\end{array}\right]$$

30. Move row 1 into column 1.

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

31. Move row 2 into column 2.

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

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

    $$\left[\begin{array}{cc}2& 0\\ -1& 3\end{array}\right]$$

33. 2 columns in the first matrix match 2 rows in the second. The product has 2 rows and 2 columns.

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

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

    $$5\times 2+1\times \left(-1\right)$$

35. Calculate this individual product, including its signs.

    $$5\times 2=10$$

36. Calculate this individual product, including its signs.

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

37. Add the calculated products to obtain this entry.

    $$10-1=9$$

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

    $$5\times 0+1\times 3$$

39. Calculate this individual product, including its signs.

    $$5\times 0=0$$

40. Calculate this individual product, including its signs.

    $$1\times 3=3$$

41. Add the calculated products to obtain this entry.

    $$0+3=3$$

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

    $$1\times 2+4\times \left(-1\right)$$

43. Calculate this individual product, including its signs.

    $$1\times 2=2$$

44. Calculate this individual product, including its signs.

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

45. Add the calculated products to obtain this entry.

    $$2-4=-2$$

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

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

47. Calculate this individual product, including its signs.

    $$1\times 0=0$$

48. Calculate this individual product, including its signs.

    $$4\times 3=12$$

49. Add the calculated products to obtain this entry.

    $$0+12=12$$

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

    $$\left[\begin{array}{cc}9& 3\\ -2& 12\end{array}\right]$$

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

    $$(\left[\begin{array}{cc}9& 3\\ -2& 12\end{array}\right],\left[\begin{array}{cc}9& 3\\ -2& 12\end{array}\right])$$

Answer

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