The document discusses matrix multiplication. It provides examples of multiplying matrices and calculating the individual elements of the resulting matrix. Matrix multiplication is only defined if the number of columns of the first matrix equals the number of rows of the second matrix. Each element of the resulting matrix is calculated by taking the inner product of the corresponding row and column from the original matrices.

1. 9.5 The Algebra of Matrices
Day Two
Ephesians 4:32 "Be kind to one another, tenderhearted,
forgiving one another, as God in Christ forgave you."

2. Matrix Multiplication
When doing A ⋅ B ,
# columns in A must = # rows in B

3. Matrix Multiplication
When doing A ⋅ B ,
# columns in A must = # rows in B
Inner Product of Row A and Column B
⎡ b1 ⎤
⎢ ⎥
b2 ⎥
⎡ a1 a2 → an ⎤ ⋅ ⎢
⎣ ⎦ ⎢ ⎥
↓ ⎥
⎢
⎢ bn ⎥
⎣ ⎦
has inner product of
a1b1 + a2b2 + ... + anbn

4. Matrix Multiplication
If A is an m x n matrix and
B is an n x k matrix
Then AB is the m x k matrix, C where cij
is the inner product of the
ith row of A and the jth column of B

5. ⎡ 5 ⎤
⎡ 5 9 2 ⎤ ⎢ ⎥
A = ⎢ ⎥ B = ⎢ −1 ⎥ Find AB
⎣ 6 5 3 ⎦ ⎢ 1 ⎥
⎣ ⎦

6. ⎡ 5 ⎤
⎡ 5 9 2 ⎤ ⎢ ⎥
A = ⎢ ⎥ B = ⎢ −1 ⎥ Find AB
⎣ 6 5 3 ⎦ ⎢ 1 ⎥
⎣ ⎦
2x3 times 3x1

7. ⎡ 5 ⎤
⎡ 5 9 2 ⎤ ⎢ ⎥
A = ⎢ ⎥ B = ⎢ −1 ⎥ Find AB
⎣ 6 5 3 ⎦ ⎢ 1 ⎥
⎣ ⎦
2x3 times 3x1
equal: we can multiply

8. ⎡ 5 ⎤
⎡ 5 9 2 ⎤ ⎢ ⎥
A = ⎢ ⎥ B = ⎢ −1 ⎥ Find AB
⎣ 6 5 3 ⎦ ⎢ 1 ⎥
⎣ ⎦
2x3 times 3x1
equal: we can multiply
dimension of product

9. ⎡ 5 ⎤
⎡ 5 9 2 ⎤ ⎢ ⎥
A = ⎢ ⎥ B = ⎢ −1 ⎥ Find AB
⎣ 6 5 3 ⎦ ⎢ 1 ⎥
⎣ ⎦
2x3 times 3x1
equal: we can multiply
dimension of product
AB11 : 5 ⋅ 5 + 9 ⋅ −1+ 2 ⋅1 → 18

10. ⎡ 5 ⎤
⎡ 5 9 2 ⎤ ⎢ ⎥
A = ⎢ ⎥ B = ⎢ −1 ⎥ Find AB
⎣ 6 5 3 ⎦ ⎢ 1 ⎥
⎣ ⎦
2x3 times 3x1
equal: we can multiply
dimension of product
AB11 : 5 ⋅ 5 + 9 ⋅ −1+ 2 ⋅1 → 18
AB21 : 6 ⋅ 5 + 5 ⋅ −1+ 3⋅1 → 28

11. ⎡ 5 ⎤
⎡ 5 9 2 ⎤ ⎢ ⎥
A = ⎢ ⎥ B = ⎢ −1 ⎥ Find AB
⎣ 6 5 3 ⎦ ⎢ 1 ⎥
⎣ ⎦
2x3 times 3x1
equal: we can multiply
dimension of product
AB11 : 5 ⋅ 5 + 9 ⋅ −1+ 2 ⋅1 → 18
AB21 : 6 ⋅ 5 + 5 ⋅ −1+ 3⋅1 → 28
⎡ 18 ⎤
AB = ⎢ ⎥
⎣ 28 ⎦
verify with calculator

12. ⎡ 2 −1 7 ⎤ ⎡ 2 5 ⎤
⎢ ⎥ ⎢ ⎥
A = ⎢ 3 0 5 ⎥ B = ⎢ 3 −2 ⎥ Find AB
⎢ −4 2 7 ⎥
⎣ ⎦ ⎢ 0 −1 ⎥
⎣ ⎦

13. ⎡ 2 −1 7 ⎤ ⎡ 2 5 ⎤
⎢ ⎥ ⎢ ⎥
A = ⎢ 3 0 5 ⎥ B = ⎢ 3 −2 ⎥ Find AB
⎢ −4 2 7 ⎥
⎣ ⎦ ⎢ 0 −1 ⎥
⎣ ⎦
3x3 times 3x2 product : 3x2

14. ⎡ 2 −1 7 ⎤ ⎡ 2 5 ⎤
⎢ ⎥ ⎢ ⎥
A = ⎢ 3 0 5 ⎥ B = ⎢ 3 −2 ⎥ Find AB
⎢ −4 2 7 ⎥
⎣ ⎦ ⎢ 0 −1 ⎥
⎣ ⎦
3x3 times 3x2 product : 3x2
AB11 : 2 ⋅ 2 + −1⋅ 3 + 7 ⋅ 0 → 1

15. ⎡ 2 −1 7 ⎤ ⎡ 2 5 ⎤
⎢ ⎥ ⎢ ⎥
A = ⎢ 3 0 5 ⎥ B = ⎢ 3 −2 ⎥ Find AB
⎢ −4 2 7 ⎥
⎣ ⎦ ⎢ 0 −1 ⎥
⎣ ⎦
3x3 times 3x2 product : 3x2
AB11 : 2 ⋅ 2 + −1⋅ 3 + 7 ⋅ 0 → 1
AB12 : 2 ⋅ 5 + −1⋅ −2 + 7 ⋅ −1 → 5

16. ⎡ 2 −1 7 ⎤ ⎡ 2 5 ⎤
⎢ ⎥ ⎢ ⎥
A = ⎢ 3 0 5 ⎥ B = ⎢ 3 −2 ⎥ Find AB
⎢ −4 2 7 ⎥
⎣ ⎦ ⎢ 0 −1 ⎥
⎣ ⎦
3x3 times 3x2 product : 3x2
AB11 : 2 ⋅ 2 + −1⋅ 3 + 7 ⋅ 0 → 1
AB12 : 2 ⋅ 5 + −1⋅ −2 + 7 ⋅ −1 → 5
AB 21 : 3⋅ 2 + 0 ⋅ 3 + 5 ⋅ 0 → 6

17. ⎡ 2 −1 7 ⎤ ⎡ 2 5 ⎤
⎢ ⎥ ⎢ ⎥
A = ⎢ 3 0 5 ⎥ B = ⎢ 3 −2 ⎥ Find AB
⎢ −4 2 7 ⎥
⎣ ⎦ ⎢ 0 −1 ⎥
⎣ ⎦
3x3 times 3x2 product : 3x2
AB11 : 2 ⋅ 2 + −1⋅ 3 + 7 ⋅ 0 → 1
AB12 : 2 ⋅ 5 + −1⋅ −2 + 7 ⋅ −1 → 5
AB 21 : 3⋅ 2 + 0 ⋅ 3 + 5 ⋅ 0 → 6
AB 22 : 3⋅ 5 + 0 ⋅ −2 + 5 ⋅ −1 → 10

18. ⎡ 2 −1 7 ⎤ ⎡ 2 5 ⎤
⎢ ⎥ ⎢ ⎥
A = ⎢ 3 0 5 ⎥ B = ⎢ 3 −2 ⎥ Find AB
⎢ −4 2 7 ⎥
⎣ ⎦ ⎢ 0 −1 ⎥
⎣ ⎦
3x3 times 3x2 product : 3x2
AB11 : 2 ⋅ 2 + −1⋅ 3 + 7 ⋅ 0 → 1
AB12 : 2 ⋅ 5 + −1⋅ −2 + 7 ⋅ −1 → 5
AB 21 : 3⋅ 2 + 0 ⋅ 3 + 5 ⋅ 0 → 6
AB 22 : 3⋅ 5 + 0 ⋅ −2 + 5 ⋅ −1 → 10
AB 31 : − 4 ⋅ 2 + 2 ⋅ 3 + 7 ⋅ 0 → − 2

19. ⎡ 2 −1 7 ⎤ ⎡ 2 5 ⎤
⎢ ⎥ ⎢ ⎥
A = ⎢ 3 0 5 ⎥ B = ⎢ 3 −2 ⎥ Find AB
⎢ −4 2 7 ⎥
⎣ ⎦ ⎢ 0 −1 ⎥
⎣ ⎦
3x3 times 3x2 product : 3x2
AB11 : 2 ⋅ 2 + −1⋅ 3 + 7 ⋅ 0 → 1
AB12 : 2 ⋅ 5 + −1⋅ −2 + 7 ⋅ −1 → 5
AB 21 : 3⋅ 2 + 0 ⋅ 3 + 5 ⋅ 0 → 6
AB 22 : 3⋅ 5 + 0 ⋅ −2 + 5 ⋅ −1 → 10
AB 31 : − 4 ⋅ 2 + 2 ⋅ 3 + 7 ⋅ 0 → − 2
AB 32 : − 4 ⋅ 5 + 2 ⋅ −2 + 7 ⋅ −1 → − 31

20. ⎡ 2 −1 7 ⎤ ⎡ 2 5 ⎤
⎢ ⎥ ⎢ ⎥
A = ⎢ 3 0 5 ⎥ B = ⎢ 3 −2 ⎥ Find AB
⎢ −4 2 7 ⎥
⎣ ⎦ ⎢ 0 −1 ⎥
⎣ ⎦
3x3 times 3x2 product : 3x2
AB11 : 2 ⋅ 2 + −1⋅ 3 + 7 ⋅ 0 → 1
AB12 : 2 ⋅ 5 + −1⋅ −2 + 7 ⋅ −1 → 5
⎡ 1 5 ⎤
AB 21 : 3⋅ 2 + 0 ⋅ 3 + 5 ⋅ 0 → 6 ⎢ ⎥
AB = ⎢ 6 10 ⎥
AB 22 : 3⋅ 5 + 0 ⋅ −2 + 5 ⋅ −1 → 10
⎢ −2 −31 ⎥
⎣ ⎦
AB 31 : − 4 ⋅ 2 + 2 ⋅ 3 + 7 ⋅ 0 → − 2
AB 32 : − 4 ⋅ 5 + 2 ⋅ −2 + 7 ⋅ −1 → − 31

21. ⎡ 2 −1 7 ⎤ ⎡ 2 5 ⎤
⎢ ⎥ ⎢ ⎥
A = ⎢ 3 0 5 ⎥ B = ⎢ 3 −2 ⎥ Find AB
⎢ −4 2 7 ⎥
⎣ ⎦ ⎢ 0 −1 ⎥
⎣ ⎦
3x3 times 3x2 product : 3x2
AB11 : 2 ⋅ 2 + −1⋅ 3 + 7 ⋅ 0 → 1
AB12 : 2 ⋅ 5 + −1⋅ −2 + 7 ⋅ −1 → 5
⎡ 1 5 ⎤
AB 21 : 3⋅ 2 + 0 ⋅ 3 + 5 ⋅ 0 → 6 ⎢ ⎥
AB = ⎢ 6 10 ⎥
AB 22 : 3⋅ 5 + 0 ⋅ −2 + 5 ⋅ −1 → 10
⎢ −2 −31 ⎥
⎣ ⎦
AB 31 : − 4 ⋅ 2 + 2 ⋅ 3 + 7 ⋅ 0 → − 2
AB 32 : − 4 ⋅ 5 + 2 ⋅ −2 + 7 ⋅ −1 → − 31
Verify with calculator. Then do BA. Does AB=BA?

22. ⎡ 2 −1 7 ⎤ ⎡ 2 5 ⎤
⎢ ⎥ ⎢ ⎥
A = ⎢ 3 0 5 ⎥ B = ⎢ 3 −2 ⎥ Find AB
⎢ −4 2 7 ⎥
⎣ ⎦ ⎢ 0 −1 ⎥
⎣ ⎦
3x3 times 3x2 product : 3x2
AB11 : 2 ⋅ 2 + −1⋅ 3 + 7 ⋅ 0 → 1
AB12 : 2 ⋅ 5 + −1⋅ −2 + 7 ⋅ −1 → 5
⎡ 1 5 ⎤
AB 21 : 3⋅ 2 + 0 ⋅ 3 + 5 ⋅ 0 → 6 ⎢ ⎥
AB = ⎢ 6 10 ⎥
AB 22 : 3⋅ 5 + 0 ⋅ −2 + 5 ⋅ −1 → 10
⎢ −2 −31 ⎥
⎣ ⎦
AB 31 : − 4 ⋅ 2 + 2 ⋅ 3 + 7 ⋅ 0 → − 2
AB 32 : − 4 ⋅ 5 + 2 ⋅ −2 + 7 ⋅ −1 → − 31
Verify with calculator. Then do BA. Does AB=BA?
Nope. Matrix Multiplication is NOT commutative ...

23. Properties for Matrix Multiplication

24. Properties for Matrix Multiplication
A ( BC ) = ( AB ) C Associative

25. Properties for Matrix Multiplication
A ( BC ) = ( AB ) C Associative
A ( B + C ) = AB + AC Distributive

26. Properties for Matrix Multiplication
A ( BC ) = ( AB ) C Associative
A ( B + C ) = AB + AC Distributive
( B + C ) A = BA + CA Distributive
but not = AB + AC

27. A Linear System written as a Matrix Equation

28. A Linear System written as a Matrix Equation
⎡ 1 1 1 ⎤ ⎡ x ⎤ ⎡ 1 ⎤
⎢ ⎥ ⎢ ⎥ ⎢ ⎥
⎢ 2 −3 0 ⎥ ⎢ y ⎥ = ⎢ 2 ⎥
⎢ 2 6 2 ⎥ ⎢ z ⎥ ⎢ 5 ⎥
⎣ ⎦ ⎣ ⎦ ⎣ ⎦

29. A Linear System written as a Matrix Equation
⎡ 1 1 1 ⎤ ⎡ x ⎤ ⎡ 1 ⎤
⎢ ⎥ ⎢ ⎥ ⎢ ⎥
⎢ 2 −3 0 ⎥ ⎢ y ⎥ = ⎢ 2 ⎥
⎢ 2 6 2 ⎥ ⎢ z ⎥ ⎢ 5 ⎥
⎣ ⎦ ⎣ ⎦ ⎣ ⎦
⎧ x + y + z = 1
⎪
⎨2x − 3y = 2
⎪2x + 6y + 2z = 5
⎩

30. A Linear System written as a Matrix Equation
⎡ 1 1 1 ⎤ ⎡ x ⎤ ⎡ 1 ⎤
⎢ ⎥ ⎢ ⎥ ⎢ ⎥
⎢ 2 −3 0 ⎥ ⎢ y ⎥ = ⎢ 2 ⎥
⎢ 2 6 2 ⎥ ⎢ z ⎥ ⎢ 5 ⎥
⎣ ⎦ ⎣ ⎦ ⎣ ⎦
⎧ x + y + z = 1
⎪
⎨2x − 3y = 2
⎪2x + 6y + 2z = 5
⎩
Read Example 7 on pp. 682, 683
Read Computer Graphics on pp. 683, 684

31. HW #11
Let us always meet each other with smile, for the smile
is the beginning of love.
Mother Teresa