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4.1 Matrix Operations
What you should learn:
Goal 1 Add and subtract matrices, multiply a matrix by a
scalar, and solve the matrix equations.
Goal 2 Use Matrices to solve real-life problems.
4.1 Matrix Operations
Vocabulary
• 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 Operations
Adding and Subtracting Matrices
2
1 + 4 0 - 6
Solution
7
To add or subtract matrices, they must have the
same dimensions.
2
Since 1 is a 3 x 1 matrix and 4 0 - 6 is a 1 x 3 matrix,
7
you cannot add them.
4.1 Matrix Operations
Adding and Subtracting Matrices
2 3 - 5 0 1 3
1 0 4 3 - 2 - 1
=
Solution
2 0 3 - 1 - 5 - 3 = 2 2 - 8
- 4 2 5
1 3 0 - (-2) 4 - (-1)
4.1 Matrix Operations
4
2 5 + 0
1
3
0
5 0
-2 1
4 - 3
54
2 - 2
4 - (-1)
- 2
7 = Cannot be done
0
4 - 6
2 - 2 =
- 1 3
0 - (-6)
1 - (-2) =
- 3 - 3
1
-4
5
6
3
- 6
4.1 Matrix Operations
Solving a Matrix Equation
Solve the matrix equation for x and y:
5
2x
4
0
2
- 1 20
8 16
10x 0
- 8x - 4x
-10x = 20
x = -2
0
8
- 4
y
2x 20
- 16x 16
-16x
Multiply by -2x
0
8
- 4
y
=y
-16(-2) = y
32 = y
4.1 Matrix Operations
Solve for x and y.
3x
2
4 9
-y
3x = 9
2 = -y
x=3
y = -2
4
4.1 Matrix Operations
Solve for x and y.
2 y
- 6
- 1 5
0 x
2y 5
- 6 x
4 - 5
8 - 7
3 - 5
8 - 7
3
8
3
8
2y + 5 = -5
-6 + x = -7
2y = -10
x = -1
y = -5
4.1 Matrix Operations
Using Matrix Operations
Write a matrix that shows the average costs in health
care from this year to next year.
Individual
$694.32
HMO Standard $451.80
HMO Plus
$489.48
Comprehensive
Individual
Comprehensive
HMO Standard
HMO Plus
$683.91
$463.10
$499.27
Family
$1725.36
$1187.76
$1248.12
This Year (A)
Family
$1699.48 Next Year (B)
$1217.45
$1273.08
4.1 Matrix Operations
Begin by adding matrix A and B to determine the
total costs for two years.
$694.32
$451.80
$489.48
$1725.36
$683.91
$1187.76 + $463.10
$499.27
$1248.12
$1378.23
$ 914.90
$ 988.75
$1699.48
$1217.45 =
$1273.08
$3424.84
$2405.21
$2521.20
4.1 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.
1 $1378.23
$ 914.90
2 $ 988.75
$689.12
$457.45
$494.38
$3424.84
$2405.21 =
$2521.20
$1712.42
$1202.61
$1260.60
4.1 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.
Multiply the matrix by .98 (1- .02) to get your reduction.
$683.91
.98 $463.10
$499.27
$670.23
$453.84
$489.28
$1699.48
$1217.45 =
$1273.08
$1665.49
$1193.10
$1247.62
4.1 Matrix Operations
Write a matrix which will show the monthly payment
following a 3% increase in the costs from matrix B.
Multiply the matrix by 1.03 to get your increase.
$683.91
1.03 $463.10
$499.27
$704.23
$476.99
$514.25
$1699.48
$1217.45 =
$1273.08
$1750.46
$1253.97
$1311.27
4.1 Matrix Operations
Reflection on the Section
What does it mean for a matrix to be a 4 x 3
matrix?
assignment
4.1 Matrix Operations
4.2 Multiplying Matrices
What you should learn:
Goal 1 Multiply two matrices
Goal 2 Use Matrix Multiplication to solve real-life
problems, such as finding the number of calories
burned.
4.2 Multiplying Matrices
Goal 1 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.
4.2 Multiplying Matrices
Finding the Product of Two Matrices
2 1
4
3
Find AB and BA if A - 2 0 and B
2
0
0 3
2 1 4 3
AB - 2 0
2
0
0 3
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 Matrices
2
AB - 2
0
2(4) 1(2)
- 2(4) 0(2)
0(4) 3(2)
1
0
3
4
2
3
0
2(3) 1(0)
- 2(3) 0(0)
0(3) 3(0)
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 Matrices
2(4) 1(2)
- 2(4) 0(2)
0(4) 3(2)
2(3) 1(0)
- 2(3) 0(0)
0(3) 3(0)
10 6
8 - 6
6
0
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 equal the number of rows in A.
4.2 Multiplying Matrices
Find the product. If it is not defined, state the reason.
2
1 4
0 1
5 2
3 4
2(1) 3(0) 4(5)
18
=
2(4) 3(1) 4(2) =
19
4.2 Multiplying Matrices
Find the product. If it is not defined, state the reason.
2 1 1 5 =
3 - 2 6 2
2(-1) 1 (6)
3(-1) - 2(6)
2(5) 1(2)
3(5) - 2(2)
4
15
=
12
11
4.2 Multiplying Matrices
Using Matrix Operations
4
If A
- 1
3 , B - 3
0
2
5 , C 1
1
0
- 2 ,
- 1
simplify each expression .
a. A(BC) 4
- 1
4
- 1
3 - 3
0 2
5 1 - 2
1 0 - 1
3 - 3
0 2
1
- 5
Multiply B by
C first!!!
Then multiply A by
the result!
4.2 Multiplying Matrices
3 - 3
0 2
5 1 - 2
1 0 - 1
4
- 1
3 - 3
0 2
1
- 5
6
3
- 11
- 1
a. A(BC) 4
- 1
4.2 Multiplying Matrices
Use the given matrices to simplify the expression.
1
1
A
2 - 1
AA
4
B
0
3
- 4
A(B+C)
AB + BC
- 3
2
C 2
1
- 3
1
- 2
3
9
1
3 - 15
15
10
- 10
- 6
4.2 Multiplying Matrices
Reflection on the Section
If A is a 3x4 matrix and B is a 2x3 matrix, which product, AB or BA, is
defined? Explain.
assignment
4.2 Multiplying Matrices
4.3 Determinants and Cramer’s Rule
What you should learn:
Goal 1 Evaluate determinants of 2x2 and 3x3 matrices.
Goal 2 Use Cramer’s Rule to solve systems of linear
equations.
4.3 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 Rule
Reflection on the Section
How do you find the determinant of a 2x2
matrix?
assignment
4.3 Determinants and Cramer’s Rule
4.4 Identity and Inverse Matrices
What you should learn:
Goal 1 Find and us inverse matrices.
Goal 2 Use Inverse Matrices to solve real-life situations.
4.4 Identity and Inverse Matrices
Reflection on the Section
How are a square matrix, its identity matrix, and the inverse matrix
related?
assignment
4.4 Identity and Inverse Matrices
4.5 Solving Systems Using Inverse Matrices
What you should learn:
Goal 1 Solve systems of linear equations using inverse
matrices
Goal 2 Use systems of linear equations to solve real-life
problems.
4.5 Solving Systems Using Inverse Matrices
Reflection on the Section
How can you use inverse matrices to solve a system of
equations?
assignment
4.5 Solving Systems Using Inverse Matrices