Holt Algebra 2

4-1,4-2 Matrices and Data

The table shows the top scores for girls in barrel racing at the 2004 National High School Rodeo finals. The data can be presented in a table or a spreadsheet as rows and columns of numbers.

Holt Algebra 2

4-1,4-2 Matrices and Data

You can also use a matrix to show table data. A matrix is a rectangular array of numbers enclosed in brackets.

Matrix A has two rows and three columns. A matrix with m rows and n columns has dimensions m n, read “m by n,” and is called an m n matrix. Matrix A has dimensions 2 3. Each value in a matrix is called an element of the matrix.

Holt Algebra 2

4-1,4-2 Matrices and Data

We name an element by its location in a matrix, where the number of the Row comes first, and then the number of the Column. This can also be expressed by using the lower case matrix letter with row and column number as subscripts.

The score 16.206 is located in row 2 column 1, so the name of this element is a2,1 (or Row 2, Column 1), and it’s value is 16.206.

Holt Algebra 2

4-1,4-2 Matrices and Data

The prices for different sandwiches are presented at right.

Example 1: Displaying Data in Matrix Form

6 in 9 in

Roast beef $3.95 $5.95

Turkey $3.75 $5.60

Tuna $3.50 $5.25

A. Display the data in matrix form. P =

3.95 5.95

3.75 5.60

3.50 5.25

B. What are the dimensions of P?

P has three rows and two columns, so it is a 3 2 matrix.

Holt Algebra 2

4-1,4-2 Matrices and Data

Example 2: Displaying Data in Matrix Form

C. What is element P3,2? What does is represent?

D. What is the name of the element 5.95?

The entry at P3,2, in row 3 column 2, is 5.25. It is the price of a 9 in. tuna sandwich.

The element 5.95 is P1,2. It is Row 1 Column 2

The prices for different sandwiches are presented at right.

6 in 9 in

Roast beef $3.95 $5.95

Turkey $3.75 $5.60

Tuna $3.50 $5.25

Holt Algebra 2

4-1,4-2 Matrices and Data

Use matrix M to answer the questions below.

Check It Out! Example 1

a. What are the dimensions of M? 3 4

11

m1,4 and m2,3

b. What is the value of m3,2?

c. The entry 0 appears for which two elements?

Holt Algebra 2

4-1,4-2 Matrices and Data

Example 3: Point Matrices

(x,y) = x

y

Points in the coordinate plane can also be represented by a matrix. This is called a Point Matrix.

Row 1 is always representing the x-coordinate, and Row 2 is always representing the y-coordinate. So any point matrix will only ever have 2 rows, since coordinates only have two values.

Holt Algebra 2

4-1,4-2 Matrices and Data

Triangle ABC is graphed in the coordinate plane on the left. Write its vertices as a point matrix.

1 3 2

-1 -1 1

A = (-1, 1)

B = (-1, 3)

C = (1, 2)

-The x-coordinates become the elements in the first row of the matrix.

X

Y

A B C

-The y-coordinates become the elements in the 2nd row of the matrix.

Example 3a: Point Matrices

Holt Algebra 2

4-1,4-2 Matrices and Data

Triangle A’B’C’ is graphed in the coordinate plane on the left. Write its vertices as a point matrix.

-1 -1 1

-1 -3 -2

A’ = (-1, -1)

B’ = (-3, -1)

C’ = (-2, 1)

-Put the x-coordinates as the elements in the first row of the matrix.

X

Y

A’ B’ C’

-Put the y-coordinates as the elements in the 2nd row of the matrix.

Example 3b: Point Matrices

Holt Algebra 2

4-1,4-2 Matrices and Data

You can add or subtract two matrices only if they have the same dimensions.

If the dimensions are the same, then you can add the corresponding elements: Those that are in the same location in each matrix (those with the same name)

Holt Algebra 2

4-1,4-2 Matrices and Data

Add or subtract, if possible.

Example 2A: Finding Matrix Sums and Differences

W + Y

Add each corresponding entry.

W = , 3 –2

1 0Y = ,

1 4

–2 3X = ,

4 7 2

5 1 –1Z =

2 –2 3

1 0 4

W + Y =

3 –2

1 0+

1 4

–2 3=

3 + 1 –2 + 4

1 + (–2) 0 + 3

4 2

–1 3=

Holt Algebra 2

4-1,4-2 Matrices and Data

X – Z

Subtract each corresponding element.

Example 2B: Finding Matrix Sums and Differences

W = , 3 –2

1 0Y = ,

1 4

–2 3X = ,

4 7 2

5 1 –1Z =

2 –2 3

1 0 4

Add or subtract, if possible.

X – Z =4 7 2

5 1 –1

2 –2 3

1 0 4–

2 9 –1

4 1 –5 =

Holt Algebra 2

4-1,4-2 Matrices and Data

X + Y

X is a 2 3 matrix, and Y is a 2 2 matrix. Because X and Y do not have the same dimensions, they cannot be added.

Example 2C: Finding Matrix Sums and Differences

W = , 3 –2

1 0Y = ,

1 4

–2 3X = ,

4 7 2

5 1 –1Z =

2 –2 3

1 0 4

Add or subtract, if possible.

Holt Algebra 2

4-1,4-2 Matrices and Data

Add or subtract if possible.

Check It Out! Example 2A

B + D

A = ,

4 –2

–3 10

2 6

B = , 4 –1 –5

3 2 8C = ,

3 2

0 –9

–5 14

D = 0 1 –3

3 0 10

Add each corresponding entry.

B + D =

+ 4 –1 –5

3 2 8

0 1 –3

3 0 10

4 + 0 –1 + 1 –5 + (–3)

3 + 3 2 + 0 8 + 10=

4 0 –8

6 2 18

Holt Algebra 2

4-1,4-2 Matrices and Data

B – A

Add or subtract if possible.

Check It Out! Example 2B

A = ,

4 –2

–3 10

2 6

B = , 4 –1 –5

3 2 8C = ,

3 2

0 –9

–5 14

D = 0 1 –3

3 0 10

B is a 2 3 matrix, and A is a 3 2 matrix. Because B and A do not have the same dimensions, they cannot be subtracted.

Holt Algebra 2

4-1,4-2 Matrices and Data

D – B Subtract corresponding entries.

Add or subtract if possible.

Check It Out! Example 2C

A = ,

4 –2

–3 10

2 6

B = , 4 –1 –5

3 2 8C = ,

3 2

0 –9

–5 14

D = 0 1 –3

3 0 10

0 1 –3

3 0 10

4 –1 –5

3 2 8–

–4 2 2

0 –2 2= D – B =

Holt Algebra 2

4-1,4-2 Matrices and Data

We already found B+D, Now find D+B

Commutative Property

A = ,

4 –2

–3 10

2 6

B = , 4 –1 –5

3 2 8C = ,

3 2

0 –9

–5 14

D = 0 1 –3

3 0 10

0 1 –3

3 0 10

4 –1 –5

3 2 8+

4 0 -8

6 2 18= D+B =

Are the results the same? Yes! So Matrix Addition is Commutative

Holt Algebra 2

4-1,4-2 Matrices and Data

B-D Subtract corresponding entries.

We already found D-B, NOW find B-D:

Commutative Property

A = ,

4 –2

–3 10

2 6

B = , 4 –1 –5

3 2 8C = ,

3 2

0 –9

–5 14

D = 0 1 –3

3 0 10

4 -1 –5

3 2 8

0 1 3

3 0 10–

4 -2 -2

0 2 -2= B-D =

Are the results the same? NO! So, Matrix Subtraction is NOT Commutative

Holt Algebra 2

4-1,4-2 Matrices and DataAssociative Property

Consider Three Matrices: A,B, and C. Assuming that they all have the same dimensions, Would changing the grouping affect the result of Matrix Addition?

Does A+(B+C) = (A+B)+C ????

YES! So, Matrix Addition is also Associative

Would changing the grouping affect the result of Matrix Subtraction?

Does A-(B-C) = (A-B)-C ????NO! So, Matrix Subtraction is also NOT Associative

Holt Algebra 2

4-1,4-2 Matrices and Data

You can multiply a matrix by a number, called a scalar. To find the product of a scalar and a matrix, or the scalar product, multiply each element by the scalar.

Holt Algebra 2

4-1,4-2 Matrices and Data

Check It Out! Example 4

Ticket Service Prices

Days Plaza Balcony

1—2 $150 $87.50

3—8 $125 $70.00

9—10 $200 $112.50

Use a scalar product to find the prices if a 20% discount is applied to the ticket service prices.

You can multiply by 0.2 and subtract from the original numbers.

150 87.5

125 70

200 112.5

– 0.2 =150 87.5

125 70

200 112.5

150 87.5

125 70

200 112.5

30 17.5

25 14

40 22.5

–

120 70

100 56

160 90

Holt Algebra 2

4-1,4-2 Matrices and Data

Example 4b: Simplifying Matrix Expressions

Evaluate 3P — Q, if possible.

P =

3 –2

1 0

2 –1

Q=4 7 2

5 1 –1R =

1 4

–2 3

0 4

P and Q do not have the same dimensions; they cannot be subtracted after the scalar products are found.

Holt Algebra 2

4-1,4-2 Matrices and Data

Example 4c: Simplifying Matrix Expressions

Evaluate 3R — P, if possible.

P =

3 –2

1 0

2 –1

Q=4 7 2

5 1 –1R =

1 4

–2 3

0 4

= 3

1 4

–2 3

0 4

–

3 –2

1 0

2 –1

=

3(1) 3(4)

3(–2) 3(3)

3(0) 3(4)

–

3 –2

1 0

2 –1

=

3 12

–6 9

0 12

–

3 –2

1 0

2 –1

0 14

–7 9

–2 13

Holt Algebra 2

4-1,4-2 Matrices and Data

Check It Out! Example 4d

Evaluate 3B + 2C, if possible.

D = [6 –3 8]A =

4 –2

–3 10C =

3 2

0 –9 B =

4 –1 –5

3 2 8

B and C do not have the same dimensions; they cannot be added after the scalar products are found.

Holt Algebra 2

4-1,4-2 Matrices and Data

Check It Out! Example 4e

D = [6 –3 8]A =

4 –2

–3 10C =

3 2

0 –9 B =

4 –1 –5

3 2 8

Evaluate 2A – 3C, if possible.

4 –2

–3 10= 2 – 3

3 2

0 –9

2(4) 2(–2)

2(–3) 2(10)= -

3(3) 3(2)

3(0) 3(–9)

8 –4

–6 20= -

9 6

0 -27 =

–1 –10

–6 47

Holt Algebra 2

4-1,4-2 Matrices and Data

= [6 –3 8] + 0.5[6 –3 8]

Check It Out! Example 4f

D = [6 –3 8]A =

4 –2

–3 10C =

3 2

0 –9 B =

4 –1 –5

3 2 8

Evaluate D + 0.5D, if possible.

= [6 –3 8] + [0.5(6) 0.5(–3) 0.5(8)]

= [6 –3 8] + [3 –1.5 4]

= [9 –4.5 12]

Holt Algebra 2

4-1,4-2 Matrices and Data

Lesson Quiz

1. What are the dimensions of A?

2. What is entry D1,2?

Evaluate if possible.

3. 10(2B + D) 4. C + 2D 5. Graph D in the

coordinate plane as ΔABC

–2

3 2

Not possibleA.

B.

C.