1. Chapter 6 – Intro to Matrices Operations/properties Now  multiplication

2. Matrix Multiplication
The product of matrices A (m x n) and B (n x p) is the m x p matrix C whose entry cij is given by:
Check to see if 2 matrices can be multiplied
- check for compatibility!
Product of matrices A and B can be calculated if :
# of columns of A equals # of rows in B

3. Matrix Multiplication Example – Sizes of Matrices and Their Product
A = 3 × 5 matrix ( # columns = 5 )
B = 5 × 3 matrix (# rows = 5)
so #columns_A = #rows_B  COMPATIBLE!
Product AB is then a 3 × 3 matrix
(#rows_A × #columns_B)
And(!) BA = 5 × 5 matrix
product of 5 × 3 and 3 × 5  5 × 5

4. Matrix Multiplication
Example – Sizes of Matrices and Their Product
Assume:
C = 3 × 5 matrix (how many columns?) 5
D = 7 × 3 matrix (how many rows?)
Are they compatible? Nope!
What about product DC ? (different order of multiplication)?

5. Matrix Multiplication
Example – Sizes of Matrices and Their Product
Now:
D = 7 × 3 matrix (how many columns?)
C = 3 × 5 matrix ( how many rows?) 3
So, product CD is undefined, but DC ~is~ defined.
Size/dimension of product DC?
product of 7 × 3 and 3 × 5  7 × 5

6. Matrix Multiplication
Example – Sizes of Matrices and Their Product
In general a m × n matrix is compatible
with a n × p matrix and product is
matrix of size m × p
[m × n ] [n × p ] = [m × p]

7. Matrix Multiplication
Example – Sizes of Matrices and Their Product
Try: Compatible or undefined? Size of product?
3 × 6 times 6 × 5 ?
6 × 5 times 3 × 6 ?
2 × 3 times 2 × 3 ?
1 × 4 times 4 × 1 ?
4 × 1 times 1 × 4 ?
bird × cat times cat × dog ?
So, product CD is undefined, but DC ~is~ defined.
Size/dimension of product DC?
DC is a 7 × 5 matrix.
In general a m × n is compatible with a n × p and product is a matrix of size m × p

8. Matrix Multiplication Example – Sizes of Matrices and Their Product
Try: Compatible or undefined? Size of product?
3 × 6 times 6 × 5 ? Yes, compatible: 3 × 5
6 × 5 times 3 × 6 ? Undefined
2 × 3 times 2 × 3 ? Undefined
1 × 4 times 4 × 1 ? Yes, compatible: 1 × 1
4 × 1 times 1 × 4 ? Yes, compatible: 4 × 4
bird × cat times cat × dog? Yes, compatible: bird × dog

9. Example – Matrix Products
× 3 times 3 × 1  1 × 1
want first row & first column
First row of matrix on left?
First column of matrix on right?
Multiply/add/combine:
C11 = 2(4) + 5(0) + 1(2) = = 10
Product: 𝟏𝟎

10. Example – Matrix Products
− × 3 times 3 × 1  2 × 1
𝑪𝟏𝟏 𝐶21 want first row & first column
First row of matrix on left?
First column of matrix on right?
Multiply/add/combine:
C11 = 1(1) + 4(-1) + 6(3) = 1 + (-4) = 15
Product: 𝟏𝟓 b. c. d.

11. Example – Matrix Products
− × 3 times 3 × 1  2 × 1
𝟏𝟓 𝐶21 want second row & first column
Second row of matrix on left?
First column of matrix on right?
Multiply/add/combine:
C21 = 2(1) + 0(-1) + 3(3) = = 11
Product: 𝟏𝟓 𝟏𝟏 b. c. d.

12. Example – Matrix Products
1 2 0 − × 3 times 3 × 2  2 × 2
𝑪𝟏𝟏 𝐶12 𝐶21 𝐶 (need four entries)
C11? First row (left matrix), first column (right):
& :
C11 = 1(1) + 2(3) + 0(5) = = 7
C12? First row (left matrix), second column(right):
& :
C12 = 1(2) + 2(4) + 0(6) = = 10

13. Example – Matrix Products
1 2 0 − × 3 times 3 × 2  2 × 2
𝟕 𝟏𝟎 𝑪𝟐𝟏 𝑪𝟐𝟐
C21? Second row (left matrix), first column (right):
& :
C21 = -2(1) + 0(3) + 5(5) = = 23
C22? Second row (left matrix), second column(right):
& :
C22 = -2(2) + 0(4) + 5(6) = = 26

14. Example – Matrix Products – Try it …!
2 5 − × 3 times 3 × 2  3 × 2
𝑪𝟏𝟏 𝑪𝟏𝟐 𝑪𝟐𝟏 𝑪𝟐𝟐 𝑪𝟑𝟏 𝑪𝟑𝟐 (need six entries)
C11? First row (left matrix), first column (right):
C11 = ?
C12? First row (left matrix), second column(right):
C12 =

15. Example – Matrix Products – Try it …!
2 5 − × 3 times 3 × 2  3 × 2
𝑪𝟏𝟏 𝑪𝟏𝟐 𝑪𝟐𝟏 𝑪𝟐𝟐 𝑪𝟑𝟏 𝑪𝟑𝟐 (need six entries)
C11? First row (left matrix), first column (right):
C11 = 2(1) + 5(3) + (-1)(5) = = 12
C12? First row (left matrix), second column(right):
C12 = 2(2) + 5(4) + (-1)(6) = – 6 = 18
𝟏𝟐 𝟏𝟖 ? ? ? ?

16. Example – Matrix Products – Try it …!
2 5 − × 3 times 3 × 2  3 × 2
𝟏𝟐 𝟏𝟖 𝑪𝟐𝟏 𝑪𝟐𝟐 𝑪𝟑𝟏 𝑪𝟑𝟐
C21? Second row (left matrix), first column (right):
C21 = ?
C22? Second row (left matrix), second column(right):
C22 =

17. Example – Matrix Products – Try it …!
2 5 − × 3 times 3 × 2  3 × 2
𝑪𝟏𝟏 𝑪𝟏𝟐 𝑪𝟐𝟏 𝑪𝟐𝟐 𝑪𝟑𝟏 𝑪𝟑𝟐
C21? Second row (left matrix), first column (right):
C21 = 1(1) + 4(3) + (0)(5) = = 13
C22? Second row (left matrix), second column(right):
C22 = 1(2) + 4(4) + (0)(6) = = 18
𝟏𝟐 𝟏𝟖 𝟏𝟑 𝟏𝟖 ? ?

18. Example – Matrix Products – Try it …!
2 5 − × 3 times 3 × 2  3 × 2
𝟏𝟐 𝟏𝟖 𝟏𝟑 𝟏𝟖 𝑪𝟑𝟏 𝑪𝟑𝟐
C31? Third row (left matrix), first column (right):
C31 = 0(1) + 3(3) + (7)(5) = = 44
C32? Third row (left matrix), second column(right):
C32 = 0(2) + 3(4) + (7)(6) = = 54
𝟏𝟐 𝟏𝟖 𝟏𝟑 𝟏𝟖 𝟒𝟒 𝟓𝟒

19. Example – Matrix Productsb. c.

20. If Note that A(BC) = (AB)C. Example – Associative Property
compute ABC in two ways.
Solution 1: 𝑨 𝑩𝑪 = 1 −2 −3 − − = 1 −2 −3 − − = −𝟒 −𝟗 −𝟏𝟖 −𝟏𝟑
Solution 2: 𝐀𝐁 𝐂= 1 −2 −3 − − = 1 −2 −5 −13 −4 − = −𝟒 −𝟗 −𝟏𝟖 −𝟏𝟑
Note that A(BC) = (AB)C.

21. Two Special Matrices (Identities)
Identity for addition: zero matrix  O
Aij = 0 for all i, j
Identity matrix for multiplication  I
Aij = 1 for all i = j Aij = 0 for all i ≠ j
(all matrix elements on diagonal are 1, all others 0)
E.g:
A+O = 𝟏 𝟐 𝟑 𝟒 + 𝟎 𝟎 𝟎 𝟎 = 𝟎 𝟎 𝟎 𝟎 + 𝟏 𝟐 𝟑 𝟒 = O+A = 𝟏 𝟐 𝟑 𝟒 = A
AI = 𝟏 𝟐 𝟑 𝟒 𝟏 𝟎 𝟎 𝟏 = 𝟏 𝟎 𝟎 𝟏 𝟏 𝟐 𝟑 𝟒 = 𝟏 𝟐 𝟑 𝟒 = A

22. Solution: Example – Matrix Operations Involving I and O If
compute each of the following.
Solution:

23. Given the price and the quantities, calculate the total cost.
Example – Cost Vector
Given the price and the quantities, calculate the total cost.
Solution:
The cost vector is

24. in matrix form by using matrix multiplication.
Example – Matrix Form of a System Using Matrix Multiplication
Write the system
in matrix form by using matrix multiplication.
Solution: If
then the single matrix equation is