Matrix- a rectangular arrangement of numbers in rows and columns

Named with capital letters [ A ] Dimensions- row x column (2 x 4)

9.2 Using Properties of Matrices

Element- each number in the matrix Named with lowercase letters and

subscripts (a₂₃) First number of subscript= row Second number of subscript=

column

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

Represent figures using matricesWrite a matrix to represent the point or polygon.

a. Point A

b. Quadrilateral ABCD

Represent figures using matricesSOLUTION

a. Point matrix for A

b. Polygon matrix for ABCD

x-coordinate

y-coordinate

–4

0

x-coordinates

y-coordinates

–4 –1 4 3

0 2 1 –1

A B C D

Adding and subtracting matrices

5 15 5

0 11 –4=

6 – 1 8 –(–7) 5 – 0

4 – 4 9 – (– 2) –1 – 3=

5 + 1 –3 + 2

6 + 3 –6 + (– 4)=

6 –1

9 –10=

5 –3

6 –6

1 2

3 –4+

a.

6 8 5

4 9 –1–

1 –7 0

4 –2 3

b.

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 =

Examples: Adding and subtracting matrices

Represent a translation using matrices

The matrix represents ∆ABC. Find the image 1 5 3

1 0 –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

3 3 3

Represent a translation using matrices

Add this to the polygon matrix for the preimage to find the image matrix.

–1 –1 –1

3 3 3

Translation matrix

+1 5 3

1 0 –1

Polygon matrix

A B C0 4 2

4 3 2=

Image matrix

A′ B′ C′

Multiplying matrices

Number of columns in matrix A= Number of rows in column B A X B = A B (mxn) x (nxp) (m x p)

equal1 0

4 5 Multiply

2 –3

–1 8.

SOLUTION

The matrices are both 2 X 2, so their product is defined. Use the following steps to find the elements of the product matrix.

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) ?=

? ?

Multiplying matrices

1 0

4 5

2 –3

–1 8

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(2) + 0(–1) 1(–3) + 0(8)=

? ?

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(2) + 0(–1) 1(–3) + 0(8)=

4(2) + 5(–1) ?

1 0

4 5 –1 8

2 –3

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.

=4(2) + 5(–1) 4(–3) + 5(8)

1(2) + 0(–1) 1(–3) + 0(8)1 0

4 5

2 –3

–1 8

Multiplying matrices

STEP 5

Simplify the product matrix.

2 –3

3 28=

4(2) + 5(–1) 4(–3) + 5(8)

1(2) + 0(–1) 1(–3) + 0(8)

Multiplying Matrices

Use the matrices below. Is the product defined? Explain.

–3

4A =

B = [2 1] C =6.7 0

–9.3 5.2

6. AB

Yes; the number of columns in A is equal to the number of rows in B.

ANSWER

Multiplying Matrices

Use the matrices below. Is the product defined? Explain.

1. BA

ANSWER

Yes; the number of columns in B is equal to the number of rows in A.

–3

4A =

B = [2 1] C =6.7 0

–9.3 5.2

Multiplying Matrices

Use the matrices below. Is the product defined? Explain.

2. AC

ANSWER

No; the number of columns in A is not equal to the number of rows in C.

–3

4A =

B = [2 1] C =6.7 0

–9.3 5.2

Examples of Multiplying Matrices

Multiply.

1 0

0 1

3 8

–4 7

SOLUTION

=0(3) + 1(–4) 0(8) + 1(7)

1(3) + 0(–4) 1(8) + 0(7)1 0

0 1

3 8

–4 7=

3 8

–4 7

Solve a real-world problemSOFTBALL

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.

Solve a real-world problem

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.

SOLUTION

Solve a real-world problem

EQUIPMENT COST TOTAL COST

Bats Balls Uniforms Dollars Dollars

Women

MenUniforms

13 42 16

15 45 18

Bats

Balls

205

40

Women

Men

?

?=

=

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 2 3 and the cost per item matrix is 3 1, so their product is a 2 1 matrix.

13 42 16

15 45 18

205

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