The document discusses the rules for matrix multiplication. It states that two matrices can only be multiplied if the number of columns of the first matrix is equal to the number of rows of the second matrix. It provides examples of multiplying different matrices and explains how to calculate each element of the resulting matrix by taking the dot product of the corresponding row and column. It also gives an example of using matrix multiplication to calculate total sales and revenue from sales data organized in matrices.

2. Learning Intention and Success
Criteria
 Learning Intention: Students will understand the
rules that define matrix multiplication and their
reasons for being
 Success Criteria: You will be determine the possibility
of multiplying two matrices by one another, and where
possible will be able to multiply a matrix by another
matrix

3. Matrix Multiplication
Requirements
 Not all matrices can be multiplied together, just as not
all matrices can be added
 The ability to multiply is dependent on the order of the
matrices
 If 𝐴 has order 𝑚 × 𝑛 (𝑚 rows and 𝑛 columns) and 𝐵
has order 𝑝 × 𝑟 (𝑝 rows and 𝑟 columns), we can
multiply 𝐴 × 𝐵 only if 𝑛 = 𝑝. The resultant matrix will
have order 𝑚 × 𝑟
 We say the product is undefined if the matrices
cannot be multiplied

4. Matrix Multiplication
Requirements
 That is, to multiply two matrices with orders
𝑚 × 𝑛 × (𝑝 × 𝑟)
 Note that 𝐴 × 𝐵 might be defined, but 𝐵 × 𝐴
undefined.
 Another way to say if multiplication is possible is: “The
number of columns in the first matrix must be equal to
the number of rows in the second matrix”
Must be the same
Order of resultant matrix

5. Examples
 If 𝐴 is a 3 × 2 matrix, 𝐵 is a 2 × 4 matrix and 𝐶 is a 3 ×
3 matrix, which of the following products will be
defined? If they are defined, what will the order of
their product be?
a) 𝐴 × 𝐵
b) 𝐵 × 𝐴
c) 𝐶 × 𝐵
d) 𝐶 × 𝐶
e) 𝐴 × 𝐴
f) 𝐵 × 𝐵 𝑇

6. Examples
 𝐴 → 3 × 2, 𝐵 → 2 × 4, 𝐶 → 3 × 3
a) 𝐴 × 𝐵
3 × 2 × 2 × 4
Can be multiplied.
Resultant order is 3 × 4
b) 𝐵 × 𝐴
2 × 4 × 3 × 2
Inside numbers do not match.
Product is undefined

7. Examples
 𝐴 → 3 × 2, 𝐵 → 2 × 4, 𝐶 → 3 × 3
c) 𝐶 × 𝐵
3 × 3 × 2 × 4
Inside numbers do not match.
Product is undefined
d) 𝐶 × 𝐶
3 × 3 × 3 × 3
Can be multiplied.
Resultant order is 3 × 3

8. Examples
 𝐴 → 3 × 2, 𝐵 → 2 × 4, 𝐶 → 3 × 3
e) 𝐴 × 𝐴
3 × 2 × 3 × 2
Inside numbers do not match.
Product is undefined
f) 𝐵 × 𝐵 𝑇
(recall that in 𝐵 𝑇
, row and columns are
swapped)
2 × 4 × 4 × 2
Inside numbers match. Can be multiplied
Resultant order is 2 × 2

9. How to multiply matrices
 Rule: Let 𝐴 and 𝐵 be matrices whose product, 𝐴 × 𝐵, is
defined as 𝐶.
 To calculate the value of element 𝑐𝑖,𝑗, we combine the 𝑖 𝑡ℎ
row of matrix 𝐴 and the 𝑗 𝑡ℎ column of matrix 𝐵.
𝑎1,1 𝑎1,2
𝑎2,1 𝑎2,2
𝑎3,1 𝑎3,2
×
𝑏1,1 𝑏1,2
𝑏2,1 𝑏2,2
=
𝑐1,1 𝑐1,2
𝑐2,1 𝑐2,2
𝑐3,1 𝑐3,2
A combination of the 2nd row of 𝐴 and the 1st column of
𝐵 gives the element in the 2nd row and 1st column of 𝐶.

10. How to multiply matrices
 How do we actually combine the elements of the row and
column?
 Consider both the row and column as a list of numbers
 Multiply the corresponding elements in each list together
 Add the results of these products together
Example:
1 2 3 ×
−2
4
0
= 1 × −2 + 2 × 4 + 3 × 0
= −2 + 8 + 0
= [6]

11. How to multiply matrices (cont)
 If 𝐶 = 𝐴 × 𝐵, then each element 𝑐𝑖,𝑗 is found by
combining row 𝑖 from matrix 𝐴 and column 𝑗 of matrix
𝐵.
Example:
1 2
3 4
×
−1 −3 −5
2 4 6
=
Row 1
Column 1

12. How to multiply matrices (cont)
 If 𝐶 = 𝐴 × 𝐵, then each element 𝑐𝑖,𝑗 is found by
combining row 𝑖 from matrix 𝐴 and column 𝑗 of matrix
𝐵.
Example:
1 2
3 4
×
−1 −3 −5
2 4 6
=
Row 1
Column 1

13. How to multiply matrices (cont)
 If 𝐶 = 𝐴 × 𝐵, then each element 𝑐𝑖,𝑗 is found by
combining row 𝑖 from matrix 𝐴 and column 𝑗 of matrix
𝐵.
Example:
1 2
3 4
×
−1 −3 −5
2 4 6
= 1 × −1 + 2 × 2
Cell 1,1
Row 1
Column 1

14. How to multiply matrices (cont)
 If 𝐶 = 𝐴 × 𝐵, then each element 𝑐𝑖,𝑗 is found by
combining row 𝑖 from matrix 𝐴 and column 𝑗 of matrix
𝐵.
Example:
1 2
3 4
×
−1 −3 −5
2 4 6
= 3Row 1
Column 2

15. How to multiply matrices (cont)
 If 𝐶 = 𝐴 × 𝐵, then each element 𝑐𝑖,𝑗 is found by
combining row 𝑖 from matrix 𝐴 and column 𝑗 of matrix
𝐵.
Example:
1 2
3 4
×
−1 −3 −5
2 4 6
= 3Row 1
Column 2

16. How to multiply matrices (cont)
 If 𝐶 = 𝐴 × 𝐵, then each element 𝑐𝑖,𝑗 is found by
combining row 𝑖 from matrix 𝐴 and column 𝑗 of matrix
𝐵.
Example:
1 2
3 4
×
−1 −3 −5
2 4 6
= 3 1 × −3 + 2 × 4Row 1
Column 2
Cell 1,2

17. How to multiply matrices (cont)
 If 𝐶 = 𝐴 × 𝐵, then each element 𝑐𝑖,𝑗 is found by
combining row 𝑖 from matrix 𝐴 and column 𝑗 of matrix
𝐵.
Example:
1 2
3 4
×
−1 −3 −5
2 4 6
= 3 5
Cell 1,3: 1 × −5 + 2 × 6
Cell 2,1: 3 × −1 + 4 × 2
Cell 2,2: 3 × −3 + 4 × 4
Cell 2,3: 3 × −5 + 4 × 6

18. How to multiply matrices (cont)
 If 𝐶 = 𝐴 × 𝐵, then each element 𝑐𝑖,𝑗 is found by
combining row 𝑖 from matrix 𝐴 and column 𝑗 of matrix
𝐵.
Example:
1 2
3 4
×
−1 −3 −5
2 4 6
= 3 5
Cell 1,3: 7
Cell 2,1: 5
Cell 2,2: 7
Cell 2,3: 9

19. How to multiply matrices (cont)
 If 𝐶 = 𝐴 × 𝐵, then each element 𝑐𝑖,𝑗 is found by
combining row 𝑖 from matrix 𝐴 and column 𝑗 of matrix
𝐵.
Example:
1 2
3 4
×
−1 −3 −5
2 4 6
=
3 5 7
5 7 9
Cell 1,3: 7
Cell 2,1: 5
Cell 2,2: 7
Cell 2,3: 9

20. Matrix Product Applications
 Juline's Noodle House sells Pad Thai for $10.40, Nasi
Goreng for $11.50 and Spring Rolls for $6.
This is represented with the cost matrix
10.40 11.50 6.00 .
 Her sales of those three items over one week are
represented in a matrix, where each column represents
a weekday, and each row represents a menu item.
5 6 4
2 1 4
12 9 7
7 15
5 9
15 21

21. Applications Example continued
a) How many Pad Thai’s were sold on Thursday
 Element at address 1,4 → 7 Pad Thais
b) Write a matrix product to calculate the amount of
money made each day, and calculate the product:
10.40 11.50 6.00 ×
5 6 4 7 15
2 1 4 5 9
12 9 7 15 21
= 147.00 127.90 129.60 220.30 385.50

22. Special Case: Multiplying By the
Identity Matrix
 When multiplying a matrix by the identity matrix
(either before or after), the original matrix does not
change.
 Ex.
1 2
3 4
×
1 0
0 1
=
1 × 1 + 2 × 0 1 × 0 + 2 × 1
3 × 1 + 4 × 0 3 × 0 + 4 × 1
= [
1 2
3 4
]