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4-3 Multiplying Matrices

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

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4-3 Multiplying Matrices

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  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 • Answer: The matrix is not defined.

  11. Multiplying Matrices • Find RS if

  12. Multiplying Matrices • Find RS if

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

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

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

  16. 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.

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