1. 9.2 Using Properties of Matrices
Matrix- a rectangular arrangement of numbers in rows and columns
Named with capital letters [ A ]
Dimensions- row x column (2 x 4)

2. Element- each number in the matrix
Named with lowercase letters and subscripts (a₂₃)
First number of subscript= row
Second number of subscript= column

3. Using Matrices to Represent Points/Figures
in Coordinate Planes
Use 2 row matrices to represent the
vertices on a 2-dimensional figure
1st row = x-coordinate
2nd row= y-coordinate
Example: Point (2,3)= 2 3

4. Represent figures using matrices
Write a matrix to represent the point or polygon.
a. Point A
b. Quadrilateral ABCD

5. Represent figures using matrices
SOLUTION
a. Point matrix for A
x-coordinate
y-coordinate –4
b. Polygon matrix for ABCD
x-coordinates
y-coordinates
– – –1
A B C D

6. Adding and subtracting matrices
5 –3
6 –6
1 2
3 –4 + a.
–3 + 2
–6 + (– 4) =
6 –1
9 –10 =
4 9 –1 –
1 –7 0
4 –2 3 b.
6 – –(–7) – 0
4 – – (– 2) –1 – 3 = –4 =

7. Examples: Adding and subtracting matrices
In Exercises 3 and 4, add or subtract. 3.
[–3 7] + [2 –5]
SOLUTION
[–3 7] + [2 –5]
=[–1 2 ] 4.
1 –4
3 –5 –
2 3
7 8
SOLUTION
1 –4
3 –5 –
2 3
7 8
–1 –7
–4 –13 =

8. Represent a translation using matrices
The matrix represents ∆ABC. Find the image –1
matrix that represents the translation of ∆ABC 1 unit left and 3 units up. Then graph ∆ABC and its image.
SOLUTION
The translation matrix is
–1 –1 –1

9. Represent a translation using matrices
Add this to the polygon matrix for the preimage to find the image matrix. + –1
Polygon matrix
A B C =
Image matrix
A′ B′ C′
–1 –1 –1
Translation matrix

10. Multiplying matrices equal Number of columns in matrix A=
Number of rows in column B
A X B = A B
(mxn) x (nxp) (m x p)
equal
1 0
4 5
2 –3
–1 8 .
Multiply
The matrices are both 2 X 2, so their product is defined. Use the following steps to find the elements of the product matrix.
SOLUTION

11. Multiplying matrices STEP 1
Multiply the numbers in the first row of the first matrix by the numbers in the first column of the second matrix. Put the result in the first row, first column of the product matrix.
1 0
4 5
2 –3
–1 8
1(2) + 0(–1) ? =
? ?

12. Multiplying matrices STEP 2
Multiply the numbers in the first row of the first matrix by the numbers in the second column of the second matrix. Put the result in the first row, second column of the product matrix.
1 0
4 5
2 –3
–1 8
1(2) + 0(–1) 1(–3) + 0(8) =
? ?

13. Multiplying matrices STEP 3
Multiply the numbers in the second row of the first matrix by the numbers in the first column of the second matrix. Put the result in the second row, first column of the product matrix.
1 0
4 5
–1 8
2 –3
1(2) + 0(–1) 1(–3) + 0(8) =
4(2) + 5(–1) ?

14. Multiplying matrices STEP 4
Multiply the numbers in the second row of the first matrix by the numbers in the second column of the second matrix. Put the result in the second row, second column of the product matrix.
1 0
4 5
2 –3
–1 8 =
4(2) + 5(–1) 4(–3) + 5(8)
1(2) + 0(–1) 1(–3) + 0(8)

15. Multiplying matrices STEP 5 Simplify the product matrix.
4(2) + 5(–1) 4(–3) + 5(8)
1(2) + 0(–1) 1(–3) + 0(8)
2 –3
3 28 =

16. Multiplying Matrices
Use the matrices below. Is the product defined? Explain. –3 4 A = C = – B
= [2 1]
6. AB
Yes; the number of columns in A is equal to the number of rows in B.
ANSWER

17. Multiplying Matrices
Use the matrices below. Is the product defined? Explain. –3 4 A = C = – B
= [2 1]
1. BA
ANSWER
Yes; the number of columns in B is equal to the number of rows in A.

18. Multiplying Matrices
Use the matrices below. Is the product defined? Explain. –3 4 A = C = – B
= [2 1]
2. AC
ANSWER
No; the number of columns in A is not equal to the number of rows in C.

19. Examples of Multiplying Matrices
1 0
3 8
0 1
–4 7
SOLUTION =
3 8
–4 7
1 0
0 1
3 8
–4 7 =
0(3) + 1(–4) 0(8) + 1(7)
1(3) + 0(–4) 1(8) + 0(7)

20. Solve a real-world problem
SOFTBALL
Two softball teams submit equipment lists for the season. A bat costs $20, a ball costs $5, and a uniform costs $40. Use matrix multiplication to find the total cost of equipment for each team.

21. Solve a real-world problem
SOLUTION
First, write the equipment lists and the costs per item in matrix form. You will use matrix multiplication, so you need to set up the matrices so that the number of columns of the equipment matrix matches the number of rows of the cost per item matrix.

22. Solve a real-world problem
EQUIPMENT COST TOTAL COST =
Bats Balls Uniforms
Dollars
Women
Men
Uniforms
Bats
Balls 20 5 40 ? =

23. Solve a real-world problem
You can find the total cost of equipment for each team by multiplying the equipment matrix by the cost per item matrix. The equipment matrix is and the cost per item matrix is , so their product is a matrix. 20 5 40
13(20) + 42(5) + 16(40) =
15(20) + 45(5) + 18(40) =
1110
1245
The total cost of equipment for the women’s team is $1110, and the total cost for the men’s team is $1245.
ANSWER