Mathematics 10 — Exercise 3.4 · Q1 (iv) AB

Full question

Find AB, if possible.

Step-by-step solution

1. 2 columns in the first matrix match 2 rows in the second. The product has 3 rows and 3 columns.

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

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

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

3. Calculate this individual product, including its signs.

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

4. Calculate this individual product, including its signs.

   $$2\times 2=4$$

5. Add the calculated products to obtain this entry.

   $$-1+4=3$$

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

   $$1\times 4+2\times 3$$

7. Calculate this individual product, including its signs.

   $$1\times 4=4$$

8. Calculate this individual product, including its signs.

   $$2\times 3=6$$

9. Add the calculated products to obtain this entry.

   $$4+6=10$$

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

    $$1\times 1+2\times 1$$

11. Calculate this individual product, including its signs.

    $$1\times 1=1$$

12. Calculate this individual product, including its signs.

    $$2\times 1=2$$

13. Add the calculated products to obtain this entry.

    $$1+2=3$$

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

    $$--1+1\times 2$$

15. Calculate this individual product, including its signs.

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

16. Calculate this individual product, including its signs.

    $$1\times 2=2$$

17. Add the calculated products to obtain this entry.

    $$1+2=3$$

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

    $$-4+1\times 3$$

19. Calculate this individual product, including its signs.

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

20. Calculate this individual product, including its signs.

    $$1\times 3=3$$

21. Add the calculated products to obtain this entry.

    $$-4+3=-1$$

22. Entry (2,3): pair row 2 with column 3 and add their products.

    $$-1+1\times 1$$

23. Calculate this individual product, including its signs.

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

24. Calculate this individual product, including its signs.

    $$1\times 1=1$$

25. Add the calculated products to obtain this entry.

    $$-1+1=0$$

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

    $$3\times \left(-1\right)+0\times 2$$

27. Calculate this individual product, including its signs.

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

28. Calculate this individual product, including its signs.

    $$0\times 2=0$$

29. Add the calculated products to obtain this entry.

    $$-3+0=-3$$

30. Entry (3,2): pair row 3 with column 2 and add their products.

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

31. Calculate this individual product, including its signs.

    $$3\times 4=12$$

32. Calculate this individual product, including its signs.

    $$0\times 3=0$$

33. Add the calculated products to obtain this entry.

    $$12+0=12$$

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

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

35. Calculate this individual product, including its signs.

    $$3\times 1=3$$

36. Calculate this individual product, including its signs.

    $$0\times 1=0$$

37. Add the calculated products to obtain this entry.

    $$3+0=3$$

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

    $$\left[\begin{array}{ccc}3& 10& 3\\ 3& -1& 0\\ -3& 12& 3\end{array}\right]$$

Answer

The final answer is shown below.