CHAPTER 4
Matrices
4.1 INTRO TO MATRICES

Matrix: a rectangular array of variables or
constants in horizontal rows and vertical columns,
usually enclosed in brackets

Element: a value in a matrix

Dimensions: number of rows x number of columns

Read “m by n”


State the dimensions of matrix G if
2 −1 0
3
G=
1 5 −3 −1
State the dimensions of matrix A if
2 6 1
A= 7 1 5
9 3 0
12 15 26
TYPES OF MATRICES

Row matrix:


7
A matrix with only one column
ex:
5
11
Square matrix:

A matrix with the same number of rows as columns


0
Column matrix:


A matrix with only one row ex: 1
ex:
4 2
9 −4
Zero matrix:

All elements are zero
SOLVE AN EQUATION INVOLVING
MATRICES
𝑦
6 − 2𝑥
1.
=
31 + 4𝑦
3𝑥
2.
5𝑥 + 2
0
𝑦−4
12 −8
=
0
2
4𝑧 + 6
NOW PLAYING
Ticket Information
3.
Evening Shows
Matinee Shows
Adult …. $7.50
Adult …. $5.50
Child ….$4.50
Child ….$4.50
Senior….$5.50
Senior ….$5.50
Twilight Shows
All tickets…….$3.75


Write a matrix for the prices of movie tickets for
adults, children, and seniors.
What are the dimensions of the matrix?
4.2 OPERATIONS WITH MATRICES

Matrices can be added and subtracted if, and only
if, they have the same dimensions.
𝑎
 ex:
𝑐
𝑒
𝑏
+
𝑔
𝑑
𝑓
𝑎+𝑒
=
ℎ
𝑐+𝑔
𝑏+𝑓
𝑑+ℎ
𝑎
 ex:
𝑐
𝑒
𝑏
−
𝑔
𝑑
𝑓
𝑎−𝑒
=
ℎ
𝑐−𝑔
𝑏−𝑓
𝑑−ℎ
1. Find A + B if A =
6
−1
4
0
and B =
−3 1
0 3
2. Find A + B if
A=
−6 7
−9 3
and B =
4 −2 0
1 5 −1
3. Find B – A if
8
A= 5
4
3
1
4
6 2
and B = 9 0
−3 0
SCALAR MULTIPLICATION


Scalar: a constant that you can multiply a
matrix by
ex:
x
𝑎
𝑐
𝑏
𝑥𝑎
=
𝑑
𝑥𝑐
𝑥𝑏
𝑥𝑑
4. Find 3A, if A =
−1 4
5 3
0 8
−7 2
5. If A =
3 7
−2 1
and B =
2 −3
, find 5A – 2B
5 −4
4.3 MULTIPLYING MATRICES

You can multiply matrices if and only if:
the number of columns in the first matrix is the
same as the number of rows in the second matrix

Ex:
A5 x 3 and B3 x 4 = AB
5x4

If the matrices cannot be multiplied =
product matrix is not defined
MULTIPLYING MATRICES
𝑤 𝑥
𝑎𝑤 + 𝑏𝑦
𝑎 𝑏
X
=
𝑦
𝑧
𝑐𝑤 + 𝑑𝑦
𝑐 𝑑
𝑎
Step 1:
𝑐
𝑤
𝑏
x 𝑦
𝑑
𝑥
𝑧
𝑎
Step 2 :
𝑐
𝑤
𝑏
x 𝑦
𝑑
𝑥
𝑧
𝑎
Step 3 :
𝑐
𝑤
𝑏
x 𝑦
𝑑
𝑥
𝑧
𝑎
Step 4 :
𝑐
𝑤
𝑏
x 𝑦
𝑑
𝑥
𝑧
𝑎𝑥 + 𝑏𝑧
𝑐𝑥 + 𝑑𝑧
FIND RS IF
1. R =
3
−1
2
−2 5
and S =
0
1 7
AT A SWIMMING MEET 6 POINTS ARE AWARDED FOR
1ST PLACE, 4 POINTS FOR 2ND PLACE, AND 3 POINTS
FOR 3RD PLACE.
2. The chart shows how many swimmers placed in each position through the meet for
the four participating schools.
School
1st Place
2nd Place
3rd Place
Central Dauphin
4
7
3
Cumberland Valley
8
8
1
Hershey
10
5
3
Carlisle
3
3
6
Write a set of matrices to model the points earned.
Which team won the meet?
COMMUTATIVE PROPERTY – DOES IT
WORK FOR MATRICES?
1 2
3. Find each product if P = 4 3 and S = 9 −3 2
6 −1 −5
0 −1
a. PS
b. SP
DISTRIBUTIVE PROPERTY – DOES IT WORK
FOR MATRICES?
4. Find each product if A = 3
−1
a. A (B + C)
1 1
2
−2 5
B=
and C =
4
−5 3
6 7
b. AB + AC
4.5 DETERMINANTS


Determinant: A number associated with a square
matrix
Second-Order Determinant

A value found by calculating the difference of the
products of the two diagonals in a 2x2 matrix

𝑎
𝑐
𝑏
= ad – bc
𝑑
FIND THE VALUE OF THE DETERMINANT
1.
−2
6
5
8
2.
7
−3
4
2
3.
9 6
0 −4
4.
6
3
12
6

Third-Order Determinant

Determinant of a 3x3 matrix

Method 1: Expansion by Minors


𝑎
𝑑
𝑔
𝑏
𝑒
ℎ
𝑐
𝑓 =a𝑒
ℎ
𝑖
Method 2: Diagonals
𝑑
𝑓
-b
𝑔
𝑖
𝑓
𝑑
+c
𝑔
𝑖
𝑒
ℎ
FIND THE DETERMINANT USING
EXPANSION BY MINORS
2 7
3. −1 5
6 9
−3
−4
0
4.6 CRAMER’S RULE


Use the determinants to solve systems of
equations
Ex: ax + by = e
cx + dy = f
x=
x=
𝑒
𝑓
𝑎
𝑐
𝑏
𝑑
𝑏
𝑑
𝑑𝑒 −𝑏𝑓
𝑎𝑑 −𝑏𝑐
and y =
and y =
𝑎 𝑒
𝑐 𝑓
𝑎 𝑏
𝑐 𝑑
𝑎𝑓 −𝑐𝑒
𝑎𝑑 −𝑏𝑐
Write the answer as
(x, y)
SOLVE THE SYSTEM OF EQUATIONS USING
CRAMER’S RULE
1. 5x + 4y = 28
3x – 2y = 8
2. 2x – 3y = 12
-6x + y = -20
IN VOTING FOR THE COLORS OF A NEW HIGH SCHOOL, BLUE & GOLD RECEIVED 440
VOTES FROM 10TH AND 11TH GRADERS WHILE RED & BLACK RECEIVED 210 VOTES FROM
THE SAME GRADES. IN THE 10TH GRADE, BLUE & GOLD RECEIVED 72% OF THE TOTAL
AND RED & BLACK RECEIVED 28%. IN THE 11TH GRADE, BLUE & GOLD RECEIVED
64% OF THE TOTAL AND RED & BLACK RECEIVED 36%.


Write a system of equations that represents the total number of
votes for each pair of colors.
Find the total number of votes cast in 10th grade and in 11th
grade.
4.7 IDENTITY AND INVERSE
MATRICES

Identity Matrix:

A square matrix that, when multiplied by another
matrix equals the same matrix

Ex:
1 0
0 1
or
1
0
0
0 0
1 0
0 1

Inverse Matrices:

When the product of two matrices with the same
dimensions is the identity matrix
DETERMINE WHETHER EACH PAIR OF
MATRICES ARE INVERSES OF EACH OTHER.
1
2
2 2
1. X =
and Y =
−1 4
−1
1
2
1
4
3 4
2. C =
1 2
and
1
D = −1
2
−2
3
2
To find the inverse of a matrix

𝑏
𝑑
Find the determinant to see if it has an inverse 𝑎𝑑 − 𝑏𝑐


𝑎
𝑐
If the determinant is zero, it cannot have an inverse
If the inverse exists it =
1
𝑎𝑑−𝑏𝑐
𝑑
−𝑐
−𝑏
𝑎
FIND THE INVERSE FOR THE GIVEN MATRIX
3. R =
−4
8
−3
6
4. A =
2
−4
1
3
4.8 USING MATRICES TO SOLVE
SYSTEMS OF EQUATIONS

Step 1: Rewrite the system of equations as a
matrix equation

Ex: 5x + 7y = 11
3x + 8y = 18
5
3

𝑥
11
7
∙ 𝑦 =
18
8
Step 2: find the inverse matrix

1
40 −21
8
−3
1
−7
8
=
19 −3
5
−7
=
5
8
19
−3
19
−7
19
5
19

Step 3: Multiply each side of the matrix equation by
the inverse matrix




8
19
−3
19
1
0
𝑥
𝑦
−7
19
5
19
0
∙
1
=
∙
5
3
𝑥
7
∙ 𝑦 =
8
𝑥
𝑦 =
8
19
−3
19
8
19
−3
19
11 +
11 +
−7
19
5
19
∙
11
18
−7
(18)
19
5
(18)
19
−38
19
57
19
Step 4: Write the solution as an ordered pair : −2, 3
SOLVE THE SYSTEM USING MATRICES
1. 5x + 3y = 13
4x + 7y = -8
2. 6a – 9b = -18
8a – 12b = 24