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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.
4.1 Matrix Operations4.1 Matrix Operations
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
4.1 Matrix Operations4.1 Matrix Operations
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
=
=
4.1 Matrix Operations4.1 Matrix Operations
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
4.1 Matrix Operations4.1 Matrix Operations
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
4.1 Matrix Operations4.1 Matrix Operations
4y - 9 4 2 3 x
3x = 9
x = 3
2 = -y
y = -2
Solve for x and y.Solve for x and y.
4.1 Matrix Operations4.1 Matrix Operations
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
4.1 Matrix Operations4.1 Matrix Operations
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
4.1 Matrix Operations4.1 Matrix Operations
$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$
=
4.1 Matrix Operations4.1 Matrix Operations
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$$1712.42 12.689$
4.1 Matrix Operations4.1 Matrix Operations
Using the matrix B on health care costs, write a matrix C for the following year that shows the costs after a 2% decrease.
$1273.08 27.499$$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$$1665.49 23.670$
4.1 Matrix Operations4.1 Matrix Operations
Write a matrix which will show the monthly payment following a 3% increase in the costs from matrix B.
$1273.08 27.499$$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$
=
4.1 Matrix Operations4.1 Matrix Operations
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
4.1 Matrix Operations4.1 Matrix Operations
4.24.2 Multiplying Matrices
What you should learn:GoalGoal 11
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
4.2 Multiplying Matrices4.2 Multiplying 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.
4.2 Multiplying Matrices4.2 Multiplying Matrices
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.
4.2 Multiplying Matrices4.2 Multiplying Matrices
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!!!
4.2 Multiplying Matrices4.2 Multiplying Matrices
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
4.2 Multiplying Matrices4.2 Multiplying Matrices
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
4.2 Multiplying Matrices4.2 Multiplying Matrices
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!
4.2 Multiplying Matrices4.2 Multiplying Matrices
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-
4.2 Multiplying Matrices4.2 Multiplying Matrices
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
4.2 Multiplying Matrices4.2 Multiplying Matrices
Reflection on the SectionReflection on the SectionReflection on the SectionReflection on the Section
If A is a 3x4 matrix and B is a 2x3 matrix, which product, AB or BA, is defined?
Explain.
assignmentassignment
4.2 Multiplying Matrices4.2 Multiplying Matrices
4.34.3 Determinants and Cramer’s Rule
What you should learn:GoalGoal 11
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.
4.3 Determinants and Cramer’s Rule4.3 Determinants and Cramer’s Rule
Reflection on the SectionReflection on the SectionReflection on the SectionReflection on the Section
How do you find the determinant of a 2x2 matrix?
assignmentassignment
4.3 Determinants and Cramer’s Rule4.3 Determinants and Cramer’s Rule
4.44.4 Identity and Inverse Matrices
What you should learn:GoalGoal 11
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
Reflection on the SectionReflection on the SectionReflection on the SectionReflection on the Section
How are a square matrix, its identity matrix, and the inverse matrix related?
assignmentassignment
4.4 Identity and Inverse Matrices4.4 Identity and Inverse Matrices
4.54.5 Solving Systems Using Inverse Matrices
What you should learn:GoalGoal 11
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
Reflection on the SectionReflection on the SectionReflection on the SectionReflection on the Section
How can you use inverse matrices to solve a system of equations?
assignmentassignment
4.5 Solving Systems Using Inverse Matrices4.5 Solving Systems Using Inverse Matrices