Matrix Multiplication

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
]