MATRICES Independent Study Developed by: B. deTreville Highland Springs High School.

MATRICES

Independent Study

Developed by: B. deTreville

Highland Springs High School

Directions

Read this PowerPoint presentation and complete assignments 1 – 4. You do not have to copy the problem for assignments 1 and 4. For assignments 2 and 3 you need to write the problem then do it.

You must clearly label each assignment.

You will earn a homework grade for each assignment. (That is FOUR homework grades!!!)

You will be quizzed on this material next class.

What is a matrix?

A matrix is an array of data. It is organized into rows and columns inside brackets.

2 4 8

6 5 1

0 9 5

Each piece of data in a matrix is called and element.

A matrix is named for its dimension. The dimension is determined by the number of rows and columns.

Writing Matrices

Now you know a matrix is an array of elements. The elements are in rows and columns. The elements are put inside brackets. The table below can be written as a matrix.

T-shirt inventory at Delmar’s Department Storepurple green yellow orange

small 10 6 3 21

medium 6 8 4 6

large 2 4 0 8

10 6 3 21

6 8 4 6

2 4 0 8

Size of Matrices: Dimension

Dimension describes the size of a matrix. The dimensions of a matrix are the number of rows by the number of columns.

rows x columns (rows by columns)

5 9 7 6

2 6 0 1

7 8 4 2

1

2

3

columns 1 2 3 4

rowsThis is a 3 x 4 (3 by 4) matrix. If we name it matrix A we would write

A3x4

Capital letters along with the dimensions are used to name a matrix.

Size of Matrices: Dimension

5 9 7 6

2 6 0 1

7 8 4 2

A3x4

How many elements does this matrix have? 12

Do you notice a relationship between the number of elements in the matrix and it’s dimensions?

If you multiply the number of rows by the number of columns the product is the number of elements in the matrix. Matrix B2x5 has 10 elements. How many elements does matrix A4x2 have? 8

Size of Matrices: Dimension

How many rows does matrix B have? 5

How many columns does matrix B have? 3

4 7 5

2.1 1 7

9 20 13

5 0 0.5

4 6 15

B =

What are the dimensions of matrix B? 5 x 3

How would you write it?

B5x3

Size of Matrices: Dimension

How many rows does matrix A have? 3

How many columns does matrix A have? 1

5

9

0

A =

What are the dimensions of matrix A? 3 x 1

How would you write it?

A3x1

Size of Matrices: Dimension

6 3

0 35

M =

What are the dimensions of matrix M? 2 x 2

How would you write it?

M2x2

Size of Matrices: Dimension

How many elements will a 3 x 7 matrix have? 21

How many elements does T6x4 have? 24

Assignment #1Tell the dimensions of each matrix.

1.A= 2.B= 3.T=

How many elements does each matrix have?

4. M3x6 5. K1x7 6. T2x2

(Continued on next slide.)

4 3

9 0

10 6.5 2.3

5 9 13

78 6 44

93 4 88

0 13 7

10 55 41

6 13 68

90 65 4

Assignment #17. Write a matrix for the following table.

Student Body at Local High School

Boys Girls

Freshman 120 152

Sophmores 115 104

Juniors 142 144

Seniors 108 117

7a. How many freshman are

there?

7b. How many students attend

Local High School?

7c. How many girls are there?

Make a matrix with each of the following dimensions.

8. 1x6 9. 2x5 10. 4x1

Matrix Addition

In order to add two matrices together, they must have the same dimensions (they must be the same size).

A = B = C=4 6

3 1

0 4

5 2

7 8 3

16 21 9

Matrices A and C can be added because they are both 2x2. Since matrix B is 2x3 it cannot be added to A or C.

Matrix Addition

Which matrices below, if any, can be added together?

A = B = C=

D =

Matrix A and D can be added because they are both 1x4.

5

8

7

2 8 15 5 7 8 9

7 8 4 1

Matrix Addition

Can matrix M and N be added?

M = N =6 3

9 6

0 7

9 14

Since they are both 2x2 they can be added.

Can matrix C and D be added?

C = D =2 0

11 15

3 7 9 34 76

17 4 33 0 15

No. They have different dimensions.

Matrix Addition

To add two or more matrices together you simply add the corresponding elements. Corresponding elements are the elements that are in the same place.

A = B=2 5

8 7

3 0

5 16

2 and -3 are corresponding elements because they are in the same position.

-5

What is the corresponding element for 7? 16

5 and 0 are corresponding elements.

What is the corresponding element for -8?

Matrix Addition

To add two or more matrices you add the corresponding elements. The result is a matrix with the same dimensions.

1

2

3

3

4

5

4

6

8

1 + 323

+

+45

Matrix Addition

Add the matrices below.

1 2 4

3 0 5

3 2 2

1 4 1

1 3 2 2 4 2

3 1 0 ( 4) 5 ( 1)

2 4 6

4 4 4

Assignment #2

A= B= C= D= 12 19 34

11 25 16

3 6 4

4 5 9

8 6

9 3

5 5

3 4

2 1

5 2

Add each pair of matrices. If the matrices cannot be added explain why.

1. C + D 2. A + B 3. A + C

4. B + A 5. A + B + A

Subtracting Matrices

To subtract two or more matrices you simply subtract the corresponding elements. The result is a matrix with the same dimensions.

1

2

3

3

4

5

2

2

2

1 - 323

-

-45

Subtracting Matrices

Subtract the matrices below.

1 2 4

3 0 5

3 2 2

1 4 1

1 3 2 2 4 ( 2)

3 1 0 ( 4) 5 ( 1)

4 0 2

2 4 6

Assignment #3

A= B= C= D= 12 19 34

11 25 16

3 6 4

4 5 9

8 6

9 3

5 5

3 4

2 1

5 2

Subtract each pair of matrices if possible.

1. C - D 2. A - B 3. A - C

4. B - A (Be careful! It’s not the same as A-B)

5. D - C (Be careful! It’s not the same as C - D)

Scalar Multiplication

Scalar multiplication is multiplying a matrix by a number. To perform scalar multiplication, simply multiply every element inside the matrix by the scalar. The resulting matrix will have the same dimensions and the original matrix.

3

2 8 10

13 4 0

22 3 7

Scalar

Scalar Multiplication

3(2) 3( 8) 3(10)

3(13) 3( 4) 3(0)

3(22) 3(3) 3(7)

6 24 30

39 12 0

66 9 21

3

2 8 10

13 4 0

22 3 7

Multiply every element in this matrix by 3.

Scalar Multiplication

2 4 1 3 Multiply every element by 2.

2(4) 2( 1) 2(3)

8 2 6

Assignment #4

Complete the following.

1. 2. 3.

4. 5. 6.

5 2 4 0

3 5 1

6

4 2 1

2 3 4

5 1 2

3

4

8 12

4 32

2.5

1 4

6 5

9 8

0 12

3

62

3

1

5

3

7

Continued on next slide.

Assignment #4

7. 8. 9.

10.

3 2 4 0

3 5 1

3

2.5

3.1

1

2

22 18

12 30

0

1 4

6 5

9 8

“Together at Last!”

Sometimes you will see scalar multiplication and matrix addition or subtraction in the same problem. NO PROBLEM! Just remember to follow the order of operations. Multiplication first then addition or subtraction.

2 3 4

6 8

( 3)

10 72

36

Do scalar multiplication first.

6 8

12 16

30 21

2 18

Then add. 36 13

10 34

Assignment #5

Complete the following. Show all work.

1. 2.

3. 4.

72 5 3

1 0 6

4

1 2 6

0 7 2

2 3 7 4 2 6

10

6.1 1.7 1.3

5 3.5 7

4.2 8 9

2

1 2

3 4

1

24 8 12 3 4 0 7

KCC First

Assignment #5

5. 6.

7. 8.

9. 10.

2 2 5 0

1 4 1

10

5 2 0

5 0 2

3 5 1 4 6 5

10

6.1 1.7 1.3

5 3.5 7

4.2 8 9

1 2 3

4 5 6

7 8 9

2

36 9 12 2 4 0 7

2 6 8 3 1 5 21 3

5 7

4 6 9 10

KCC First

KCC First

The End Is Near!

Congratulations. You have completed all the assignments for this independent study.