1. 4-3 Multiplying Matrices
Objectives:
Multiply matrices.
Use the properties of matrix multiplication.

2. Multiplying Matrices
You can multiply two matrices if and only if the number of columns in the first matrix is equal to the number of rows in the second matrix.
When you multiplying two matrices Amxn and Bnxr, the resulting matrix AB is an m x r matrix.

3. Dimensions of Matrix Products
Determine whether each matrix product is defined. If so, state the dimensions of the product.
Example: A4x6 and B6x2

4. Dimensions of Matrix Products
Determine whether each matrix product is defined. If so, state the dimensions of the product.
Example: A4x6 and B6x2
Answer: 4x2

5. Dimensions of Matrix Products
Determine whether each matrix product is defined. If so, state the dimensions of the product.
Example: A4x6 and B6x2
Answer: 4x2
Example: A3x4 and B4x2

6. Dimensions of Matrix Products
Determine whether each matrix product is defined. If so, state the dimensions of the product.
Example: A4x6 and B6x2
Answer: 4x2
Example: A3x4 and B4x2
Answer: 3x2

7. Dimensions of Matrix Products
Determine whether each matrix product is defined. If so, state the dimensions of the product.
Example: A3x2 and B3x2

8. Dimensions of Matrix Products
Determine whether each matrix product is defined. If so, state the dimensions of the product.
Example: A3x2 and B3x2
Answer: The matrix is not defined.

9. Dimensions of Matrix Products
Determine whether each matrix product is defined. If so, state the dimensions of the product.
Example: A3x2 and B3x2
Answer: The matrix is not defined.
Example: A3x2 and B4x3

10. Dimensions of Matrix Products
Determine whether each matrix product is defined. If so, state the dimensions of the product.
Example: A3x2 and B3x2
Answer: The matrix is not defined.
Example: A3x2 and B4x3

11. Multiplying Matrices
Find RS if

12. Multiplying Matrices
Find RS if

13. (first row, first column)
Multiplying Matrices
Find RS if
(first row, first column)

14. (first row, second column)
Multiplying Matrices
Find RS if
(first row, second column)

15. (second row, first column)
Multiplying Matrices
Find RS if
(second row, first column)

16. (second row, second column)
Multiplying Matrices
Find RS if
(second row, second column)

17. Multiplying Matrices
Find UV if

18. Multiplying Matrices
Find UV if

19. Multiplying Matrices
Find UV if

20. Multiplying Matrices
Find UV if

21. Multiplying Matrices
Find UV if

22. Properties of Multiplying Matrices
Matrix multiplication is NOT commutative.
This means that if A and B are matrices, AB≠BA.

23. AB≠BA in Matrices
Find KL if

24. AB≠BA in Matrices
Find KL if

25. AB≠BA in Matrices
Find KL if

26. AB≠BA in Matrices
Find KL if

27. AB≠BA in Matrices
Find KL if

28. AB≠BA in Matrices
Find LK if

29. AB≠BA in Matrices
Find LK if

30. AB≠BA in Matrices
Find LK if

31. AB≠BA in Matrices
Find LK if

32. AB≠BA in Matrices As you can see, multiplication is NOT commutative.
The order of multiplication matters.

33. Properties of Multiplying Matrices
Distributive Property
If A, B, and C are matrices, then
A(B+C)=AB+AC and
(B+C)A=BA+CA

34. Distributive Property
Find A(B+C) if

35. Distributive Property
Find A(B+C) if

36. Distributive Property
Find A(B+C) if

37. Distributive Property
Find A(B+C) if

38. Distributive Property
Find A(B+C) if

39. Distributive Property
Find A(B+C) if

40. Distributive Property
Find A(B+C) if

41. Distributive Property
Find AB+AC if

42. Distributive Property
Find AB+AC if

43. Distributive Property
Find AB+AC if

44. Distributive Property
Find AB+AC if

45. Distributive Property
Find AB+AC if

46. Distributive Property
Find AB+AC if

47. Distributive Property
Find AB+AC if

48. Distributive Property
As you can see, you can extend the distributive property to matrices.