Matrices - Multiplication of Matrices

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- The document discusses how to multiply matrices, including defining when multiplication is valid based on the number of columns and rows of the matrices. - It provides an example of multiplying two matrices, showing the step-by-step process of multiplying corresponding elements and summing them. - It also discusses properties of matrix multiplication, including associativity, distributivity, and the existence of an identity matrix. - Examples are provided to illustrate verifying these properties by calculating different matrix multiplications.

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Matrices - Multiplication of Matrices

1. How do we multiply two matrices? Learning Objective

2. To calculate multiplication of matrices To interpret properties of multiplication To validate properties of multiplication Learning Outcomes

3. Matrices Let's multiply two matrices! What does it mean?  The product of two matrices A and B is defined if the number of columns of A is equal to the number of rows of B.

4. Matrices Let's multiply two matrices! What does it mean?  The product of two matrices A and B is defined if the number of columns of A is equal to the number of rows of B.  Let A = [aij] be an m × n matrix and B = [bjk] be an n × p matrix.

5. Matrices Let's multiply two matrices! What does it mean?  The product of two matrices A and B is defined if the number of columns of A is equal to the number of rows of B.  Let A = [aij] be an m × n matrix and B = [bjk] be an n × p matrix. Then the product of the matrices A and B is the matrix C of order m × p.

6. Matrices Let's multiply two matrices! Let's take an example!  The product of two matrices A and B is defined if the number of columns of A is equal to the number of rows of B.  Let A = [aij] be an m × n matrix and B = [bjk] be an n × p matrix. Then the product of the matrices A and B is the matrix C of order m × p.  To get the (i, k)th element cik of the matrix C, we take the ith row of A and kth column of B, multiply them elementwise and take the sum of all these products.

7. Matrices Let's multiply two matrices!  If C= , D = .

8. Matrices Let's multiply two matrices!  If C= , D = , then CD =

9. Matrices Let's multiply two matrices! Let's illustrate it!  If C= , D = , then CD = This is a 2 × 2 matrix in which each entry is the sum of the products across some row of C with the corresponding entries down some column of D.

10. Matrices Let's multiply two matrices!  If C= , D = , then CD =

11. Matrices Let's multiply two matrices!  If C= , D = , then CD =

12. Matrices Let's multiply two matrices!  If C= , D = , then CD =

13. Matrices Let's multiply two matrices!  If C= , D = , then CD =

14. Matrices Let's multiply two matrices!  If C= , D = , then CD =

15. Matrices Let's multiply two matrices!  If C= , D = , then CD =

16. Matrices Let's multiply two matrices!  If C= , D = , then CD =

17. Matrices Let's multiply two matrices!  If C= , D = , then CD =

18. Matrices Let's multiply two matrices!  If C= , D = , then CD = =

19. To calculate multiplication of matrices To interpret properties of multiplication To validate properties of multiplication Learning Outcomes How confident do you feel?

20. To calculate multiplication of matrices To interpret properties of multiplication To validate properties of multiplication Learning Outcomes How confident do you feel?

21.  The multiplication of matrices possesses the following properties, which we state without proof. Matrices Let's multiply two matrices!

22.  The multiplication of matrices possesses the following properties, which we state without proof. 1. The associative law: For any three matrices A, B and C. We have (AB) C = A (BC), whenever both sides of the equality are defined. Matrices Let's multiply two matrices! What does it mean?

23.  The multiplication of matrices possesses the following properties, which we state without proof. 1. The associative law: For any three matrices A, B and C. We have (AB) C = A (BC), whenever both sides of the equality are defined. 2. The distributive law: For three matrices A, B and C. (i) A (B+C) = AB + AC Matrices Let's multiply two matrices! What does it mean?

24.  The multiplication of matrices possesses the following properties, which we state without proof. 1. The associative law: For any three matrices A, B and C. We have (AB) C = A (BC), whenever both sides of the equality are defined. 2. The distributive law: For three matrices A, B and C. (i) A (B+C) = AB + AC (ii) (A+B) C = AC + BC, whenever both sides of equality are defined. Matrices Let's multiply two matrices! What does it mean?

25.  The multiplication of matrices possesses the following properties, which we state without proof. 1. The associative law: For any three matrices A, B and C. We have (AB) C = A (BC), whenever both sides of the equality are defined. 2. The distributive law: For three matrices A, B and C. (i) A (B+C) = AB + AC (ii) (A+B) C = AC + BC, whenever both sides of equality are defined. 3. The existence of multiplicative identity: For every square matrix A, there exist an identity matrix of same order such that IA = AI = A. Matrices Let's multiply two matrices!

26. To calculate multiplication of matrices To interpret properties of multiplication To validate properties of multiplication Learning Outcomes How confident do you feel?

27. To calculate multiplication of matrices To interpret properties of multiplication To validate properties of multiplication Learning Outcomes How confident do you feel?

28.  If A = , B = and C = ,find A(BC), (AB)C and show that (AB)C = A(BC). Matrices Let's verify the properties! Let's solve it!

29.  If A = , B = and C = . Calculate AC, BC and (A + B)C. Also, verify that (A + B)C = AC + BC. Matrices Let's verify the properties! Let's solve it!

30. Matrices Verify that (A + B)C = AC + BC.

31. Matrices Verify that (A + B)C = AC + BC.

32. Matrices Verify that (A + B)C = AC + BC.

33. Matrices Verify that (A + B)C = AC + BC.

34. Matrices Verify that (A + B)C = AC + BC.

35. Matrices Verify that (A + B)C = AC + BC.

36. Matrices Verify that (A + B)C = AC + BC.

37. Matrices Verify that (A + B)C = AC + BC.

38. Matrices Verify that (A + B)C = AC + BC.

39. Matrices Verify that (A + B)C = AC + BC.

40. Matrices Verify that (A + B)C = AC + BC.

41. Matrices Verify that (A + B)C = AC + BC.

42. Matrices Verify that (A + B)C = AC + BC.

43. Matrices Verify that (A + B)C = AC + BC.

44. Matrices Verify that (A + B)C = AC + BC.

45. Matrices Verify that (A + B)C = AC + BC.

46. Matrices Verify that (A + B)C = AC + BC.

47. Matrices Verify that (A + B)C = AC + BC.

48. To calculate multiplication of matrices To interpret properties of multiplication To validate properties of multiplication Learning Outcomes How confident do you feel?

49. To calculate multiplication of matrices To interpret properties of multiplication To validate properties of multiplication Learning Outcomes How confident do you feel?

50. How do we multiply two matrices? Learning Objective

51. Matrices Learning Activity In a legislative assembly election, a political group hired a public relations firm to promote its candidate in three ways: telephone, house calls, and letters. The cost per contact (in paise) is given in matrix A as: The number of contacts of each type made in two cities X and Y is given by: Find the total amount spent by the group in the two cities X and Y.

Editor's Notes

#2: An overview of the content of the lesson Must be in the form of a question where appropriate Students should be able to answer the question at the end - either fully, partly or in a way that demonstrates they understand what gaps in their knowledge they need to address Verbs such as to understand / to know / to gain confidence / to learn Ask students to give the question a go and point out that, at the end of the lesson, they should be able to answer fully
#3: Measurable outcomes that students can demonstrate and self-assess against Must be written using Bloom’s taxonomy verbs Verbs based on students ability and pitch of lesson It must be clear that students understand the outcomes before moving on Make an activity of this slide: Ask students to read this aloud Ask them to paraphrase Ask that they explain what they mean Ask what they already know related to these outcomes There may be as few as 2 outcomes, or max 4
#4: The next slides should be focused on achieving first outcome Make reference to the outcome in the teaching Fill this with thinking skills activities, peer assessment, higher-order questioning, engaging activities and challenge
#5: The next slides should be focused on achieving first outcome Make reference to the outcome in the teaching Fill this with thinking skills activities, peer assessment, higher-order questioning, engaging activities and challenge
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#18: The next slides should be focused on achieving first outcome Make reference to the outcome in the teaching Fill this with thinking skills activities, peer assessment, higher-order questioning, engaging activities and challenge
#19: The next slides should be focused on achieving first outcome Make reference to the outcome in the teaching Fill this with thinking skills activities, peer assessment, higher-order questioning, engaging activities and challenge
#20: Revisit the first outcome and use the polling function to allow students to privately self-assess You may feel that the students do not need privacy to self-assess and in this instance, the chat box may be used Polling must be used until you can fully assess their confidence to use the chat box and express honesty If students self-assess as a 4/5, ensure that you are fully confident in their assessment Ask questions Ask for examples Students to ask each other questions If a few students self-assesses as a 3, but others as a 4/5, discretely ask the higher ones to give examples and to explain their achievement/understanding If all students are a 3 or below, do not move on. Move to a blank page at the end of the presentation and use as a whiteboard to further explain If students are ½, go back to the beginning Always ask students what the gaps are and help them to identify these in order to promote metacognition
#21: 1. The outcome changes colour when achieved to the same colour as the objective to demonstrate the connection, progress and what happens next
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#27: As previously.
#28: As previously.
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#49: As previously.
#50: As previously.
#51: An overview of the content of the lesson Must be in the form of a question where appropriate Students should be able to answer the question at the end - either fully, partly or in a way that demonstrates they understand what gaps in their knowledge they need to address Verbs such as to understand / to know / to gain confidence / to learn Ask students to give the question a go and point out that, at the end of the lesson, they should be able to answer fully
#52: The next slides should be focused on achieving first outcome Make reference to the outcome in the teaching Fill this with thinking skills activities, peer assessment, higher-order questioning, engaging activities and challenge

Matrices - Multiplication of Matrices

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Editor's Notes