4.1 Matrix Operations What you should learn: Goal1 Goal2 Add and subtract matrices, multiply a...

4.14.1 Matrix Operations

What you should learn:GoalGoal 11

GoalGoal 22

Add and subtract matrices, multiply a matrix by a scalar, and solve the matrix equations.

Use Matrices to solve real-life problems.

4.1 Matrix Operations4.1 Matrix Operations

VocabularyVocabulary• A matrix is a rectangular arrangement of

numbers in rows and columns where the numbers are called entries.

• The dimensions of a matrix are given as the number of rows x the number of columns.

• Scalar multiplication is the process of multiplying each entry in a matrix by a scalar, a real number.


Adding and Subtracting MatricesAdding and Subtracting Matrices

+Solution

To add or subtract matrices, they must have the same dimensions.

Since is a 3 x 1 matrix and is a 1 x 3 matrix,

you cannot add them.

7 12

7 12

6- 0 4

6- 0 4


4 0 1

5- 3 2

1- 2- 3 3 1 0

Solution

(-1)-4 (-2)- 0 3 13 -5- 1 -3 02

5 2 4- 8- 2 2

=

=


Adding and Subtracting MatricesAdding and Subtracting Matrices

3- 1

5 2

0 07 02- 4

3-3- (-1)-4 (-2)-1 2-2(-6)-0 45

3- 4 1 2- 0 5

=+ Cannot be done

3 1- 2- 2 6- 4

=

=

6- 5 3 4- 6 1


Solving a Matrix Equation

Solve the matrix equation for x and y:

y 8 16

4- 0 20 8 2 41- 0 52x

16x-4x -8x - 2x 0 10x

y 8 16

4- 0 20

Multiply by -2x

-10x = 20

x = -2-16x = y

-16(-2) = y

32 = y


4y - 9 4 2 3 x

3x = 9

x = 3

2 = -y

y = -2

Solve for x and y.Solve for x and y.


Solve for x and y.Solve for x and y.

8 7-3 5- 8x

4 5 0 6-1- 2y

8 7-3 5- 8 x 6-

3 5 2y

2y + 5 = -5

2y = -10

y = -5

-6 + x = -7

x = -1


Using Matrix Operations

Write a matrix that shows the average costs in health care from this year to next year.

$1248.12 48.489$$1187.76 80.451$$1725.36 32.694$Comprehensive

HMO Standard

HMO Plus

$1273.08 27.499$$1217.45 10.463$$1699.48 91.683$

This Year (A)

Next Year (B)Comprehensive

HMO Standard

HMO Plus

Individual Family

Individual Family


$1248.12 48.489$$1187.76 80.451$$1725.36 32.694$

$1273.08 27.499$$1217.45 10.463$$1699.48 91.683$

Begin by adding matrix A and B to determine the total costs for two years.

+

$2521.20 988.75 $$2405.21 914.90 $$3424.84 23.1378$

=


Multiply the result by ½, which is equivalent to dividing by 2. Round your answers to the nearest cent to find the average.

$2521.20 988.75 $$2405.21 914.90 $$3424.84 23.1378$

2

1 =

$1260.60 $494.38$1202.61 457.45$$1712.42 12.689$


Using the matrix B on health care costs, write a matrix C for the following year that shows the costs after a 2% decrease.

$1273.08 27.499$$1217.45 10.463$$1699.48 91.683$

98.

Multiply the matrix by .98 (1- .02) to get your reduction.

=

$1247.62 28.489$$1193.10 84.453$$1665.49 23.670$


Write a matrix which will show the monthly payment following a 3% increase in the costs from matrix B.

$1273.08 27.499$$1217.45 10.463$$1699.48 91.683$

03.1

Multiply the matrix by 1.03 to get your increase.

$1311.27 25.514$$1253.97 99.476$$1750.46 23.704$

=


Reflection on the SectionReflection on the SectionReflection on the SectionReflection on the Section

What does it mean for a matrix to be a 4 x 3 matrix?

assignmentassignment


4.24.2 Multiplying Matrices


GoalGoal 22

Multiply two matrices

Use Matrix Multiplication to solve real-life problems, such as finding the number of calories burned.

4.2 Multiplying Matrices4.2 Multiplying Matrices

GoalGoal 11 Multiply two matrices


When multiplying two matrices A times B, the number of columns in A must equal the number of rows B.

If A is an m x n matrix and B is an n x p matrix, then the product AB is an m x p matrix.

Finding the Product of Two Matrices

0 2

3 4 B and 3 0 0 2-1 2

A ifBA and AB Find

0 2

3 4 3 0 0 2-1 2

AB

Because the number of columns in A equals the number of rows in B, the product is defined. AB will be a 3 x 2 matrix.


0 2

3 4 3 0 0 2-1 2

AB

3(0) 0(3) 3(2) 0(4) 0(0) 2(3)- 0(2) 2(4)-

1(0) 2(3) 1(2) 2(4)

Multiply corresponding entries in the first row of A and the first column of B. Then add. Use similar procedure to write the other entries.


3(0) 0(3) 3(2) 0(4) 0(0) 2(3)- 0(2) 2(4)-

1(0) 2(3) 1(2) 2(4)

0 6 6- 86 10

BA is undefined because B is a 2 x 2 matrix and A is a 3 x 2 matrix. The number of columns in B does not equaldoes not equal the number of rows in A.

YES! Order is

importa

nt!!!


Find the product. If it is not defined, state the reason.

4 3 2

4(2) 3(1) 2(4) 4(5) 3(0) )1(2 =

=

19 18


2 5

1 0

4 1

Find the product. If it is not defined, state the reason.

2 6 5 1 2- 3

1 2

2(2)- 3(5) 2(6)- 3(-1)1(2) 2(5) (6) 1 (-1)2

=

=

11 15

12 4


Using Matrix Operations

.expressioneach simplify

, 1- 02- 1C , 1 2

5 3- B , 0 1-3 4 A

If

1- 0

2- 1 1 2 5 3- 0 1-

3 4 A(BC) .aMultiply B by C first!!!

5- 2

1 3- 0 1-3 4

Then multiply A by the result!


1- 0

2- 1 1 2 5 3- 0 1-

3 4 A(BC) .a

5- 2

1 3- 0 1-3 4

1- 3

11- 6-


1- 2

1 1A

Use the given matrices to simplify the expression.

2 0

3- 4 B

1 1

3- 2C

AA

3 4-2- 3

A(B+C)

15- 3 9 1

AB + BC

6- 10 10- 15



If A is a 3x4 matrix and B is a 2x3 matrix, which product, AB or BA, is defined?

Explain.



4.34.3 Determinants and Cramer’s Rule


GoalGoal 22

Evaluate determinants of 2x2 and 3x3 matrices.

Use Cramer’s Rule to solve systems of linear equations.

4.3 Determinants and Cramer’s Rule4.3 Determinants and Cramer’s Rule

The determinant of a square matrix A is denoted by det A or |A|.

Cramer’s Rule is a method of solving a system of linear equations using the determinant of the coefficient matrix of the linear system.

The entries in the coefficient matrix are the coefficients of the variables in the same order.



How do you find the determinant of a 2x2 matrix?



4.44.4 Identity and Inverse Matrices


GoalGoal 22

Find and us inverse matrices.

Use Inverse Matrices to solve real-life situations.

4.4 Identity and Inverse Matrices4.4 Identity and Inverse Matrices


How are a square matrix, its identity matrix, and the inverse matrix related?


4.4 Identity and Inverse Matrices4.4 Identity and Inverse Matrices

4.54.5 Solving Systems Using Inverse Matrices


GoalGoal 22

Solve systems of linear equations using inverse matrices

Use systems of linear equations to solve real-life problems.

4.5 Solving Systems Using Inverse Matrices4.5 Solving Systems Using Inverse Matrices


How can you use inverse matrices to solve a system of equations?


4.5 Solving Systems Using Inverse Matrices4.5 Solving Systems Using Inverse Matrices

4.1 Matrix Operations What you should learn: Goal1 Goal2 Add and subtract matrices, multiply a...

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