The document discusses using matrix multiplication to manipulate matrices. It provides an example of a matrix representing the number of TVs sold at different stores each day of the week. It shows how to use matrix multiplication to calculate the total number of TVs sold each week at each store and each day across all stores. The document also discusses using diagonal matrices to change parts of other matrices, like increasing one store's prices by 30% and giving another store a 50% discount.

2. Learning Intention and Success
Criteria
 Learning Intention: Students will understand how to
use matrix multiplication to manipulate matrices
 Success Criteria: You will be able sum the rows or
columns of a matrix, and multiply individual rows or
columns using matrix multiplication

3. Summation Matrices
 Often, it can be useful to add up the elements in each
row or column of a matrix.
 For example, if a matrix shows the number of items
sold each day at a variety of different stores, you may
want to create a matrix that shows the number of
total number of items sold at each store in a week.
 The secret lies in a list of 1’s

4. Adding rows
 Adding up rows:
 To add up the rows of a matrix 𝐴, we multiply 𝐴 by 𝐶, a
column matrix of 1’s (𝐴 × 𝐶)
 Adding up columns:
 To add up the columns of a matrix 𝐵, we multiply 𝑅, a
row matrix of 1’s, by 𝐵 (𝑅 × 𝐵)

5. Example
 Example: The matrix below represents the number of
TVs sold at three different electronic stores over a week
form Monday to Friday
𝑆 =
2 0 1 2 5
3 1 1 4 4
1 0 0 1 3
𝑆𝑡𝑜𝑟𝑒 𝐴
𝑆𝑡𝑜𝑟𝑒 𝐵
𝑆𝑡𝑜𝑟𝑒 𝐶
 How many TVs were sold at Store A on Wednesday?
 𝑠1,3 = 1 TV

6. Example Continued
𝑆 =
2 0 1 2 5
3 1 1 4 4
1 0 0 1 3
𝑆𝑡𝑜𝑟𝑒 𝐴
𝑆𝑡𝑜𝑟𝑒 𝐵
𝑆𝑡𝑜𝑟𝑒 𝐶
 Write a matrix expression that will determine the number of TVs
sold all week at each location
2 0 1 2 5
3 1 1 4 4
1 0 0 1 3
×
1
1
1
1
1
=
2 × 1 + 0 × 1 + 1 × 1 + 2 × 1 + 5 × 1

7. Example Continued
𝑆 =
2 0 1 2 5
3 1 1 4 4
1 0 0 1 3
𝑆𝑡𝑜𝑟𝑒 𝐴
𝑆𝑡𝑜𝑟𝑒 𝐵
𝑆𝑡𝑜𝑟𝑒 𝐶
 Write a matrix expression that will determine the number of TVs
sold all week at each location
2 0 1 2 5
3 1 1 4 4
1 0 0 1 3
×
1
1
1
1
1
=
10
13
5

8. Example Continued
𝑆 =
2 0 1 2 5
3 1 1 4 4
1 0 0 1 3
𝑆𝑡𝑜𝑟𝑒 𝐴
𝑆𝑡𝑜𝑟𝑒 𝐵
𝑆𝑡𝑜𝑟𝑒 𝐶
 Now, write a matrix calculation that will determine the number
of TVs sold each day of the week in total across the three stores.
1 1 1 ×
2 0 1 2 5
3 1 1 4 4
1 0 0 1 3
= 1 × 2 + 1 × 3 + 1 × 1

9. Example Continued
𝑆 =
2 0 1 2 5
3 1 1 4 4
1 0 0 1 3
𝑆𝑡𝑜𝑟𝑒 𝐴
𝑆𝑡𝑜𝑟𝑒 𝐵
𝑆𝑡𝑜𝑟𝑒 𝐶
 Now, write a matrix calculation that will determine the number
of TVs sold each day of the week in total across the three stores.
1 1 1 ×
2 0 1 2 5
3 1 1 4 4
1 0 0 1 3
= 6 1 2 7 12

10. Changing Part of a Matrix
 Motivation: The matrix below represents the cost of
salad, chips and drinks at three local cafes.
𝑆𝑎𝑙𝑎𝑑 𝐶ℎ𝑖𝑝𝑠 𝐷𝑟𝑖𝑛𝑘
14.50 5.40 4.80
12.80 6.30 5.00
15.00 4.50 3.30
𝐶𝑎𝑓𝑒 𝐴
𝐶𝑎𝑓𝑒 𝐵
𝐶𝑎𝑓𝑒 𝐶
 Say Café A increases their all their prices by 30%. How
can we deal with this using matrix multiplication?

11. Changing Part of a Matrix
 We multiply by a diagonal matrix, 𝐷.
 The numbers along the leading diagonals represent
the multiplication factors for specific rows or columns
 For 𝐷 × 𝐴, we will be changing the rows of A
 For 𝐴 × 𝐷, we change the columns of A.

12. Example revisited
𝐴 =
𝑆𝑎𝑙𝑎𝑑 𝐶ℎ𝑖𝑝𝑠 𝐷𝑟𝑖𝑛𝑘
14.50 5.40 4.80
12.80 6.30 5.00
15.00 4.50 3.30
𝐶𝑎𝑓𝑒 𝐴
𝐶𝑎𝑓𝑒 𝐵
𝐶𝑎𝑓𝑒 𝐶
 Café A increases their all their prices by 30%, and Café
C has a half price sale. Show this using matrix
multiplication.
 Increase by 30%  multiply by 1.3
 Decrease by 50%  multiply by 0.5

13. Example revisited
𝐴 =
𝑆𝑎𝑙𝑎𝑑 𝐶ℎ𝑖𝑝𝑠 𝐷𝑟𝑖𝑛𝑘
14.50 5.40 4.80
12.80 6.30 5.00
15.00 4.50 3.30
𝐶𝑎𝑓𝑒 𝐴
𝐶𝑎𝑓𝑒 𝐵
𝐶𝑎𝑓𝑒 𝐶
 Café A increases their all their prices by 30%, and Café
C has a half price sale. Show this using matrix
multiplication.
𝐷 =
1.30 0 0
0 1 0
0 0 0.50

14. Example revisited
𝐷 × 𝐴 =
1.30 0 0
0 1 0
0 0 0.50
×
14.50 5.40 4.80
12.80 6.30 5.00
15.00 4.50 3.30
=
18.85 7.02 6.24
12.80 6.30 5.00
7.50 4.25 1.65