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Chapter 4
Matrices
In Chapter 4, You Will…
• Move from using matrices in organizing
data to manipulating matrices through
data.
• Learn to represent real-world
relationships by writing matrices and using
operations such as addition and
multiplication to develop new matrices.
4-1 Organizing Data
What you’ll learn …
• To identify matrices and their elements
• To organize data into matrices
1.04 Operate with matrices
to model and solve problems.
A matrix (plural matrices) is a
rectangular array of numbers written
within brackets.
The number of horizontal rows and
the number of vertical columns
determine the dimensions of a
matrix.
Columns
Rows
Example 1 Writing the
Dimensions of a Matrix
Write the dimensions of each matrix.
Each number in a matrix is a matrix
element. You can identify a matrix
element by its position within the
matrix. Use a lowercase letter with
subscripts. The subscripts represent
the element’s row number and column
number.
Consider the matrix
The element a21 = 1, since the element in the 2nd row and 1st column is 1.
The element a13 = 9, since the element in the 1st row and 3rd column is 9.
Example 2 Identifying a Matrix Element
Identify each matrix element
17
10.4
9
a. a33
24
12
30
b. a11
3
15
15
c. a21
d. a12
Example 4a Real World Example
Gymnast
Floor
Exercise
Vault
Balance
Beam
Uneven
Bars
Amy Chow
9.525
9.468
9.625
9.400
Dominique
Dawes
9.087
9.393
8.600
9.675
Kristin
Maloney
9.525
9.225
9.312
9.575
Elise Ray
9.225
9.468
9.687
9.687
•
Which element
represents
Maloney’s score on
the vault?
• Write a matrix W to represent the information.
Example 4b Real World Example
4000
3900
3400
3500
3000
2500
Greece
Canada
USA
2000
1500
1000
500
0
400
1985-1989
50
100
1990-1994
25
Example 4b Real World Example Continued
• Write a matrix M to represent the data in
the graph, with columns representing
years.
• What are the dimensions of this matrix?
• What does the first row represent?
• What does m32 represent?
4-2 Adding and Subtracting Matrices
What you’ll learn …
• To add and subtract matrices.
• To solve certain matrix equations.
1.04 Operate with matrices to model
and solve problems.
Adding and Subtracting Matrices
You must perform matrix addition or subtraction
on matrices with equal dimensions by adding or
subtracting the corresponding elements, which
are elements in the same position in each matrix.
Example
1 -2 0
3 -5 7
-12 24
-3 5
-1 10
Adding Matrices
3 9 -3
-9 6 12
-3 1
2 -4
-1 5
The additive identity matrix for the set of
all m x n matrices is the zero matrix 0, or
Omxn ,whose elements are all zeros. The
opposite, or additive inverse, of an m x n
matrix A is –A. -A is the m x n matrix with
elements that are the opposites of the
corresponding elements of A.
A=
2 -4
-1 0
A + (-A) = 0
A+0=A
-A =
-2 4
1 0
Example 2
Using Identity and Inverse Matrices
1 2
5 -7
0 0
0 0
2 8
-3 0
-2 -8
3 0
Properties of Matrix Addition
If A, B, and C are m x n matrices, then
• A + B is an m x n matrix
Closure Property
• A+B=B+A
Commutative Property of Addition
• (A+B)+C = A+(B+C) Associative Property of Addition
• There exist a unique m x n matrix O such that
O+A=A+O=A.
Additive Identity Property
• For each A, there exists a unique opposite, -A.
A+(-A)=0
Additive Inverse Property
Subtracting Matrices
1 -2 0
3 -5 7
3 9 -3
-9 6 12
Just Add the Opposite
1 -2 0
3 -5 7
-3 -9 3
9 -6 -12
Example 3
Subtracting Matrices
6 -9 7
-2 1 8
-4 3 0
6 5 10
-3 5
-1 -10
-3 1
2 -4
A matrix equation is an equation in which the
variable is a matrix. You can use the addition and
subtraction properties of equality to solve matrix
equations.
X
+
-1 0
2 5
10
7
=
-4 4
Equal matrices are matrices with the
same dimensions and equal corresponding
elements.
Example 4 Solving A Matrix Equation
• Solve X - 1 1
= 0 1
• Solve X + -1 0
= 10 7
2 5
-4 4
3
2
8 9
Example 5 Determining Equal Matrices
Determine whether the two matrices in each pair are equal.
-0.75 1/5
½ -2
4
6
8
-3/4 0.2
0.5 -2
8/2 18/3 16/2
Example 6 Finding Unknown Matrix Elements
2x-5 4
3 3y+12
x+8 -5
3 -y
25 4
3 y+18
38 -5
3 4y-10
X = ____
Y = ____
X = ____
Y = ____
4-3
Matrix Multiplication
What you’ll learn …
• To multiply a matrix by a scalar
• To multiply two matrices
1.04 Operate with matrices to model and
solve problems.
You can multiply a matrix by a real
number.
3 5
=
3
2 8
9 15
6 24
The real number factor (such as 3) is called a
scalar.
You find the scalar product by multiplying each
element of the matrix by the scalar.
Example Scalar Multiplication
15 -12 10 0
-3
20 -10 7 0
Example 2 Using Scalar Products
2
3
-7
A=
1 4 5
Find 5B- 4A
3 0 6
B=
-1 8 2
Find A + 6B
Properties of Scalar Multiplication
If A, B, and O are m x n matrices and c and d
are scalars, then
• cA is an m x n matrix
Closure property
• (cd)A = c(dA)
Associative Property of
Multiplication
• C(A+B) = cA+cB
(c+d)A = cA + cb
• 1 (A) = A
• 0(A) = c0 = 0
Distributive Property
Multiplication Identity Property
Multiplication Property of 0
Example 3a Solving Matrix Equations with Scalars
3 4
4x + 2
-2 1
=
10 0
4 2
Example 3b Solving Matrix Equations with Scalars
-3x +
7 0 -1
2 -3 4
=
10
0 8
-19 -18 10
Investigation: Using Matrices
1.
2.
3.
How much money did the cafeteria collect selling lunch
1? Selling Lunch 2? Selling Lunch 3?
a. How much did the cafeteria collect selling all 3
lunches?
b. Explain how you used the data in the table to find
your answer.
a. Write a 1x3 matrix to represent the cost of the
lunches.
b. Write a 1x 3 matrix to represent the number of
lunches sold.
c. Describe a procedure for using your matrices to
find how much money the cafeteria collected from selling
all three lunches.
Lunch
1
Lunch 2
Lunch 3
Cost per
Lunch
$2.50
$1.75
$2.00
Number Sold
50
100
75
To perform matrix
multiplication, multiply the
elements of each row of
the first matrix by the
elements of each column of
the second matrix. Add
the products.
**Multiply rows times columns.
**You can only multiply if the number of columns in
the 1st matrix is equal to the number of rows in the
2nd matrix.
8 2 
3 2 5  


1
5
7 1 0  


 
0 3

They must match.
Dimensions:
2x3
3x2
The dimensions of your answer.
**Multiply rows times columns.
**You can only multiply if the number of
columns in the 1st matrix is equal to the
number of rows in the 2nd matrix
3 9 2  2 1
2. 



5 7 6   3 4 
Dimensions: 2 x 3
2x2
*They don’t match so can’t be
multiplied together.*
0 -1
1 0
*Answer should be a 2 x 2
4 -3
-2 5
2x2
2x2
0(4) + (-1)(-2)
0(-3) + (-1)(5)
1(4) + 0(-2)
1(-3) +0(5)
2 -5
 

4 -3
Example:
-2 5
3 -1
-2(4)+5(2)
4 -4
2 6
-2(-4)+5(6)
3(4) + -1(2) 3(-4) + -1(6)
2
38
10 -18
Example 4a
Multiplying Matrices
• Find the product of
-1 0
3 -4
-3 3
5 0
• Multiply a11 and b11. Add the products.
• The result is the element in the first row
and first column.
• Repeat with the rest of the rows and
columns.
Example 4b
Multiplying Matrices
• Find the product of
-3 3
5 0
-1 0
3 -4
• Multiply a11 and b11. Add the products.
• The result is the element in the first row
and first column.
• Repeat with the rest of the rows and
columns.
Example 5 Real World Connection
A used-record store sells tapes, LP records,
and compact discs. The matrices show
today’s information. Find the store’s gross
income for the day.
Tapes
$8
LPs
$6
CDs
$13
Tapes
LPs
CDs
9
30
20
Example 5
Find each product.
12
3
10
-5
10
-5
12 3
0 0
Example 6a Determining When a
Product Matrix Exists
8 0
2 3
Use matrices G = -1 8 and H = 2 -5 .
4 0
Determine whether products GH and HG are
defined (exist) or undefined (do not exist).
Example 6b Determining When a
Product Matrix Exists
Use matrices R = 4 -2 and S =
5 -4
8 0 -1 0
2 -5 . 1 8
Determine whether products RS and SR are
defined (exist) or undefined (do not exist).
Properties of Matrix Multiplication
If A, B, and C are n x n matrices, then
• AB is an n x n
Closure Property
• (AB)C = A(BC)
•
• A(B+C) = AB + BC
(B+C)A = BA + CA
Associative Property of
Multiplication
Distributive Property
• OA = AO = O, where O has the same
dimensions as A.
Multiplicative Property of 0
4-5 2x2 Matrices, Determinants, and Inverses
What you’ll learn …
• To evaluate determinants of 2x2
matrices and find inverse matrices
• To use inverse matrices in solving matrix
equations
1.04 Operate with matrices to model
and solve problems.
Evaluating Determinants of 2x2 Matrices
•
A square matrix is a matrix with the same
number of columns as rows.
• For an n x n square matrix, the multiplicative
identity matrix is an n x n square matrix I, with
1’s along the diagonal and zeros elsewhere.
Multiplicative Inverse of a Matrix
If A and X are n x n matrices, and
AX=XA=I, then X is the multiplicative
inverse of A, written A-1.
A
-1
(A )
=
-1
A (A)
=I
Example 1 Verifying Inverses
A=
2 3
1 2
3 -1
M=
7 -1
B=
2 -3
-1 2
N = .1 .1
-.7 .3
Determinant of a 2x2 Matrix
The determinant of a 2x2 matrix
is ad – bc.
a b
c d
-5(-3) – 7(2)
15 – 14
1
Example 2 Evaluating the Determinants
of a 2x2 Matrix
4 2
det
4 2
det
8 7
det 2 3
k
3
3-k -3
Inverse of a 2x2 Matrix
Let A =
a
b
c d
. If det A ≠0 then A has an inverse.
If det A ≠0, then A-1
=
1__ __1__
=
=
detA
ad - bc
d -b
-c a
Example 3
Finding an Inverse Matrix
2 4
1 3
12
9
4
3
.5 2.3
3 7.2
6 5
25 20
Example 4
Solving a Matrix Equation
-2 -5 X = -2
2
1
3
3
4
-4
-5
0 -22
X =
0 -28
AX = B
A-1AX = A-1B
IX = A-1B
X= A-1B
4-6
3x3 Matrices, Determinants,
and Inverses
What you’ll learn …
• To evaluate determinants of 3x3
matrices
1.04 Operate with matrices to model and
solve problems.
Matrix Determinants
 A Determinant is a real number
associated with a matrix. Only
SQUARE matrices have a determinant.
 The symbol for a determinant can be
the phrase “det” in front of a matrix
variable, det(A); or vertical bars
around a matrix, |A| or 3 1 .
2 4
Matrix Determinants
To find the determinant of a 3 x 3 matrix, first
recopy the first two columns. Then do 6 diagonal
products.
18 60 16
5 2 6 5 2
2 1 4 2 1
3 3 4 3 3
-20 -24 36
94 - (-8)
94 +8
= 102
Step 1: Rewrite first two
columns of the matrix.
Step 2: multiply diagonals
going up!
Step 3: multiply diagonals
going down!
Step 4: Bottom minus top!
Example 1 Evaluating the Determinant of a
3x3 Matrix
-1
2
0
3
-4
1
5
6
-1
Step 1: Rewrite first
two columns of the
matrix.
Step 2: multiply
diagonals going up!
Step 3: multiply
diagonals going down!
Step 4: Bottom minus
top!
Now You Are Asking …
How can I do this on the
calculator?
• Step 1: Go to Matrix (above the x-1 key)
• Step 2: Arrow to the right to EDIT to allow for
entering the matrix.
• Step 3: Type in the dimensions (size) of your matrix
and enter the values (press ENTER).
• Step 4: Go to Matrix again. Notice that your
matrix (3x3) now appears
• showing it in residence.
• Step 5:
Arrow to the right to MATH. Choose #1: det (
• Step 6: The function det( will appear on the home screen
waiting for a parameter (in this case, the
name of the determinant to evaluate).
• Step 7: Go to Matrix again. Choose [A] or whichever
location holds your matrix.
• Step 8: The name of the determinant's location will appear
as the
parameter.
Hit ENTER to see the evaluation.
Example 2 Using a Calculator
-1 3
2 -4
0 1
5
6
-1
1
0
3
2
2
10
-1
4
-6
Step 1: Enter matrix A
into your calculator.
Step 2: Use the matrix
submenus to evaluate
the determinant of A.
Identity Matrices


An identity matrix is a square matrix
that has 1’s along the main diagonal and
0’s everywhere else.
1 0 0 
1 0

0 1 0 
0 1 




 0 0 1 
When you multiply a matrix by the
identity matrix, you get the original
matrix.
Inverse Matrices

When you multiply a matrix and its
inverse, you get the identity matrix.
If A and B are inverse matrices, then AB=BA=I.
Matrix A
Inverse of
Matrix A
Identity
 3 1  2 1   1 0 
X
 5 2   5 3   0 1 


 

Example 4 Solving a Matrix Equation
0
1
1
0
3
-2
2
0
1
0
1
0
2
-2
1
1
4
0
X =
X =
0
-6
19
-1
8
-2
Use the equation
X = A-1C
4-7
Inverse Matrices and Systems
What you’ll learn …
• To solve systems of equations using inverse
matrices.
1.04 Operate with matrices to model and solve
problems.
2.10 Use systems of two or more equations or
inequalities to model and solve problems; justify
results. Solve using tables, graphs, matrix operations,
and algebraic properties.
You can represent a system of equations with a
matrix equation.
System of Equation
x + 2y = 5
3x +5y = 14
Matrix Equation
1
3
2
5
x
5
y = 14
Example 1 and 2 Writing and Solving a System as
a Matrix.
System of
Equation
5a + 3b = 7
3a + 2b = 5
System of
Equation
x + 3y = 22
3x + 2y = 10
=
=
Example 3 Solving a System of Three Equations.
System of
Equation
2x + y +3z = 1
5x + y – 2z = 8
x – y -9z = 5
System of
Equation
x–y+z=0
x – 2y – z = 5
2x - y + 2z = 8
=
=
Example 5
Unique Solutions.
System of Equation
x+y=3
x–y=7
System of
Equation
3x + 5y = 1
2x –y = -8
Remember…..
a b = ad - bc
det
c d
In Chapter 4, You Should Have…
• Moved from using matrices in
organizing data to manipulating
matrices through data.
• Learned to represent real-world
relationships by writing matrices and
using operations such as addition and
multiplication to develop new
matrices.