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Multiply Matrices

Multiply Matrices. Objectives: To multiply matrices. Warm-up. How do you find the dimensions of a matrix? We’ve learned how to add and subtract matrices and how to multiply by a scalar. How do you think we should multiply matrices?. Objective 1. You will be able to multiply matrices.

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Multiply Matrices

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  1. Multiply Matrices Objectives: • To multiply matrices

  2. Warm-up • How do you find the dimensions of a matrix? • We’ve learned how to add and subtract matrices and how to multiply by a scalar. How do you think we should multiply matrices?

  3. Objective 1 You will be able to multiply matrices

  4. Matrix Multiplication Multiplying two matrices is weird. Thank you, Arthur Cayley (1858). Because of the way it is defined, if we are multiplying two matrices , then the number of columns in must match the number of rows in . Arthur Cayley, 1850ish

  5. Matrix Multiplication If and , Must match Order: then . Arthur Cayley, 1850ish Order:

  6. Getting Rid of the Middle Man In general, if matrix has order and has order , then has order . When multiplying two matrices, the first thing you should do is find the order of the product!

  7. Exercise 1 Can you find ? If so, what is the order of the product? • : ; : • : ; :

  8. Exercise 2a Multiply.

  9. Exercise 2b Multiply.

  10. Exercise 2c Multiply.

  11. Exercise 2d Multiply.

  12. Exercise 2d Multiply.

  13. Exercise 2d Multiply.

  14. Exercise 2d Multiply.

  15. Exercise 2d Multiply.

  16. Exercise 2d Multiply.

  17. Matrix Multiplication Definition of Matrix Multiplication This definition will make more sense once we do an application question. Column Rowx

  18. Matrix Multiplication Definition of Matrix Multiplication Column +Ro +ol +ou +wm +wn = RC RRooww x

  19. The Order of the Product So you might be asking why the number of columns of the first matrix must equal the number of rows in the second matrix. This is because we must multiply a row (in the first matrix) by a column (in the second matrix), and the number of elements must match. How many elements are there in any row in the first matrix? How many elements are there in any column in the second matrix?

  20. Exercise 3a Two softball teams submit equipment lists to their sponsors. If a bat costs $21, a ball costs $4, and a glove costs $30, use matrices to find the total cost of equipment for each team.

  21. Exercise 3b The previous Exercise simplifies to the following matrix multiplication problem. Can you now see why matrix multiplication is so weird?

  22. Exercise 4 , , and

  23. Exercise 5 , , and

  24. Properties of Matrix Multiplication Let , , and be matrices and let be a scalar. Remember: There is no commutative property for matrix multiplication!

  25. Identity Matrix Recall that multiplying a number by 1 gives you back the same number. This is the Identity Property of Multiplication. Similarly, multiplying a matrix by the identity matrix, , will return the original matrix.

  26. Identity Matrix The identity matrix has 1s along the main diagonal and zeros everywhere else.

  27. Exercise 6 What is the identity matrix for a ?

  28. Exercise 7 Find .

  29. Exercise 8 Find .

  30. 3.6: Multiply Matrices Objectives: • To multiply matrices

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