MATRICES

(CHAPTER 7)

2

MATRICES

The dimensions of a matrix are written number of rows by number of columns.

In 1 – 3 give the dimensions of the matrix.

1)

823

472 2)

529

534

972

851

3)

10

12

Each number in the brackets is called an element of the matrix. In example 2 above, the

4 is the element in row 3 column 1.

4) According to the Census Bureau, in 1900 the median age at first marriage was 25.9

for males and 21.9 for females. In 1930, the median age at first marriage was 24.3

for males and 21.3 for females. In 1960, it was 22.8 for males and 20.3 for females. In

1990, it was 26.1 for males and 23.9 for females. Store this information in a 2 x 4

matrix.

5) The matrix at the right gives data from the

1992 Statistical Abstract of the United States

regarding prisoners executed under civil

authority from 1930 to 1989.

a) Give the dimensions of this matrix.

b) What does the entry in row 2, column 1 represent?

c) How many people were executed in the period 1970 – 1979?

3

MATRIX ADDITION AND SUBTRACTION Definition: If two matrices A and B have the same dimensions, their sum A + B is the

matrix in which each element is the sum of the corresponding elements in A and B.

Example 1: Example 2:

141812

182015

643

875

+

534

465

321

423

=

987

654

321

+

987

654

321

=

Definition: If two matrices A and B have the same dimensions, their difference A - B is the

matrix in which each element of B is subtracted from the corresponding element in A.

Example 1: Example 2:

141812

182015

643

875

-

534

465

321

423

=

534

465

321

423

-

141812

182015

643

875

=

SCALAR MULTIPLICATION

Matrix addition is related to a special multiplication involving matrices called scalar

multiplication. Consider this repeated addition:

24

87 +

24

87 +

24

87 =

Notice that in the final result, every element of the original matrix has been multiplied by 3.

With real numbers, we use multiplication as a shorthand for repeated addition; for

example, the sum 7 + 7 + 7 can be written as 3 x 7. Similarly, we can rewrite the above

sum as:

24

873

The constant 3 is called a scalar. Scalar Multiplication is the product of a scalar k and a

matrix A is the matrix kA in which each element is k times the corresponding element in A.

Find the product of:

1194

1275 =

4

USING MATRICES TO SOLVE SYSTEMS

Write the following systems of equations as a matrix:

1)

20y10x8

9y4x3 2)

8yx

36zyx5

11z7y4x3

Write a system of equations for each and then write the system as a matrix:

3) At a local video rental store John rents two movies and three games for a total of

$15.50. At the same time, Meg rents three movies and one game for a total of

$12.05. How much money is needed to rent a combination of one game and one

movie?

4) The total attendance at a school play was 1, 250. The cost of tickets was $6 for

students and $7.50 for adults. The school drama club had a revenue total of

$8, 362.50. How many of each ticket was sold for the play?

5) Joe bought two new compact disks and 3 used compact disks for $54. At the same

prices, Susan bought three new compact disks and one used compact disk for $53.

Find the cost of buying a new and a used compact disk.

6) Yolanda, Raphael, and Tim go to Bonzo Burger. Yolanda gets 1 burger, 1 fry, 2

sodas (she’s thirsty) and her total is $6.90. Raphael gets 2 burgers, 2 fries, 1 soda and

spends $10. Tim gets 3 burgers, 1 fry, and 1 soda, which totals $10.70. What is the

price of one soda?

5

Using x, y and z (when necessary) as your variables write a system of equations given the

following matrices:

1)

1973

2814

8423

2)

1413

228

3)

410

221

4) Using number 3 above what is the value of y?

Find x.

5)

3100

6210

9531

6) Using number 5 above what is the value of z?

Find x and y.

6

USING ROW-ECHELON FORM TO SOLVE MATRICES

A matrix in Row-Echelon form when:

All rows consisting entirely of zero in the bottom matrix.

For each row that does not consist entirely of zeros, the first non-zero entry is

1.(leading 1)

For two successive (nonzero) rows the leading 1 in the higher row is further to the left

than the leading 1 in the lower row.

Determine if the following matrices are in row-echelon form:

1)

1973

2814

8423

2)

410

221

3)

0000

6210

9531

4)

1413

228

5)

3100

6210

9532

In order to get a matrix in row-echelon form you need to apply row operations.

1) Interchange any two rows. ex. R1 R2

2) Replace a row by a nonzero multiple of that row. ex. 2R1

3) Replace a row by the sum of that row and a constant nonzero multiple of

another row. ex. –2R1 + R3

Use row operations to put the following matrix in row-echelon form:

3125

2242

7

1) Solve the system:

17z5y5x2

4y3x

9z3y2x

2) Solve the system:

12y6x3

16y4x2

3) Solve the system:

1zy2x3

0z2y2x2

4z2y3x5

8

4) Solve the system:

3zx3

8zyx2

9z3y2x

5) Solve the system:

1z3yx2

0zy2x

1zy2x3

9

6) John, Paul, and Mike go to Burger Land. John gets 5 burgers, 2 fries and 3 apple

pies for a total of $14.50. Paul gets 3 burgers, 1 fry and 2 apple pie for $8.50. Mike

gets 1 Burger, 1 fry and 1 apple pie which totals $4.50 (what a light-weight!). What is

the price of one burger?

7) Solve the system:

19wz7y4x

2w3zy4x2

2zy2x

3w2zy

10

8) Solve the system:

3wzy2x5

3w6z2y5x

1wy4x3

6w2zyx2

11

NUMBER OF SOLUTIONS

The number of Solutions of a Linear System:

For a system of linear equations, exactly one of the following is true:

1) There is exactly one solution.

2) There are infinitely many solutions.

3) There is no solution.

A system of linear equations is called consistent if it has at least one solution. A consistent

system with exactly one solution is independent. A consistent system with infinitely many

solutions is dependent. A system of linear equations is called inconsistent if it has no

solutions.

Directions: For the following matrices determine whether they are consistent or

inconsistent. If it is consistent determine if it is independent or dependent.

1)

4000

2810

8421

2)

410

221

3)

0000

6210

9531

4)

1411

000

5)

3100

6210

9531

6)

323

700

12

Solve the following systems:

1)

1z3y2x

2z2yx2

1zy3x

2)

1y2x

0zy

1z3yx

13

3)

0zx3

3zyx2

6zyx

4)

4yx4

8z2y3x

2zyx

14

MATRIX MULTIPLICATION Row By Column Multiplication:

In linear-combination applications, it is useful to store data in matrices.

Consider a movie theater that charges $6 for adults over 17, $4 for students 13 – 17, and

$2.50 for children 12 or under. How much does it cost a family with 2 adults, 1 student, and

3 children to enter the theater?

The answer is $62 + $41 + $2.53 = $23.50. This is the same arithmetic needed to

calculate the product of these matrices:

3

1

2

5.246

cost per category number in each category

Multiplying Two Matrices

In general, the product AB or AB of two matrices A and B is found by multiplying the rows

of matrix A by the columns of matrix B.

**Matrices can only be multiplied when the number of columns for the left matrix equals

the number of rows for the right matrix.

These matrices can be multiplied: These matrices cannot be

multiplied:

14

28

240

531 =

22164

36168

240

531

14

28

2 x 2 2 x 3 2 x 3 2 x 3 2 x 2

equal not equal

dimensions of product

In general, multiplication of matrices is not commutative.

Example:

3

1

2

5.246 =

Example: Let A =

14

28 and B =

240

531. Find the product AB.

15

1) 1102

0

1

0

7

0

1

1

5

2

0

1

1

2)

0

2

0

1

0

2

1

3

4

213

3)

1021

1102

0

1

0

7

0

1

1

5

2

0

1

1

4)

31

12

10

11

5) Suppose you sell 3 T-shirts at $10 each, 4 hats at $15 each, and 1 pair of shorts at

$20. (Set up 2 matrices in order to find the total revenue.)

16

IDENTITY MATRIX

1) What number is the multiplicative identity? Why?

2) Fill in the matrix to make the following statement true:

a)

24

87

??

?? =

24

87

b)

987

654

321

???

???

???

=

987

654

321

Definition of Identity Matrix:

The n x n matrix that consists of 1’s on its main diagonal and 0’s elsewhere is called the

identity matrix of order n and is denoted by

10000

0...100

0...010

0...001

In

If A is an n x n matrix then A nI =A.

17

INVERSE OF A SQUARE MATRIX

1) Solve for x without using division and show work: 5x = 10

2) What is the multiplicative inverse of 5? Why?

Definition of the Inverse of a Square Matrix:

Let A be an n x n matrix and let In be the n x n identity matrix. If there exists a matrix A-1

such that

AA-1 = In = A-1A

then A-1 is called the inverse of A.

4) Show that B is the inverse of A, where A =

11

21 and B =

11

21

To find the inverse of a 2 x 2 matrix:

The inverse of a 2 x 2 matrix

dc

ba is

ac

bd

bcad

1. If ad – bc = 0, the inverse does

not exist.

Find the inverse of the following matrices:

5)

22

13

18

6)

62

31

7)

31

41

8)

11

12

9)

563

342

211

19

SOLVING A 2 X 2 USING THE INVERSE

We can use inverses to solve a system of linear equations.

If A is an invertible matrix, the system of linear equations represented by AX = B has a

unique solution

X = A 1 B

Ex) Use an inverse matrix to solve the following systems.

1)

10y3x2

5y2x

2)

2y3x2

4y2x

20

DETERMINANTS

Determinant:

Every square matrix can be associated with a real number called its determinant. A

determinant can be positive, negative or zero. A zero determinant tells us that the lines

have the same slope and the inverse does not exist. If the determinant equals zero, the

system of linear equations has either no solution or infinitely many solutions.

A determinant can be denoted as either det(A) or |A|.

If 2 x 2 matrix A =

dc

ba , the det(A) = ad – bc.

Find the determinant of each matrix.

a)

21

32

b)

24

12

c)

422

30

To find the determinant of a square matrix of order 3 x 3 or higher, we must find what are

called minors and cofactors.

If A =

503

342

210

det(A)=

21

Find the determinant of the following matrices.

A =

871

425

321

B =

036

451

372

C =

432

761

305

22

CRAMER’S RULE

To solve a system of linear equations using Cramer’s Rule, x = det

det x , y = det

det y,…

Just watch. You’ll get it.

Use Cramer’s Rule to solve the following systems:

1)

11y5x3

10y2x4

2)

4y3x5

2y4x3

3)

1z6y2x5

10z3y2x2

5zyx4

23

CRYPTOGRAPHY

A cryptogram is a message written according to a secret code. Matrix multiplication can

be used to encode and decode messages. To begin, you need to assign a number to

each letter in the alphabet (with 0 assigned to a blank space).

To encode a message, choose an n x n invertible matrix A and multiply the un-coded row

matrices by A (on the right) to obtain coded row matrices.

Use the following matrix to encode the message MEET ME MONDAY.

A =

411

311

221

To decode the message, we simply multiply the coded row matrices by A 1 (on the right).