What is a matrix?
• When do Matrices come up in
the real world?
• Business?
• Education?
• Any time you have to organize
numbers in a table you are using
a matrix.
Matrices
• Interpret data in a matrix.
• What does the 80 mean?
• What does the 9 mean?
• Can I change the order of the rows
And still have the same
information?
• Can I change the order of the
columns and still have the same
information?
Average Price of a Hotel in given city
Phoenix Salt Lake
Moab
LA
January
100
90
100
100
February
100
90
110
100
March
140
90
110
100
April
140
95
140
100
May
100
95
130
100
June
100
100
130
150
Special matrices
• Square matrix
3 −1 5
2 4
•
or 1 3 3
1 7
2 3 6
• Row matrix
• 5 11
or 2
−3
• Column matrix
•
2
3
5
3
• 1
4
• Recognize and use matrix
notation.
• An m x n matrix gives use the
number of rows (m) and the
number of columns (n)
• Give the dimensions of each
matrix.
3
•
4
2
3
5
3
2
2 3
•
0 −1
1
2
3
8
3
4 2
9 1
6 3
Matrices
• Recognize and use matrix
notation.
Matrices
• Organize data in a matrix.
• Suppose you are a junior high
teacher. You are starting the
year by making packets for your
students. You have 7th grade
packets chapter 1, 6 pages
chapter 2, 8 pages, chapter 3, 6
pages. You have 8th grade
packets chapter 1, 4 pages,
chapter 2, 10 pages, chapter 3, 6
pages.
• Organize data in a matrix
• A businessman wants to do a
company excursion.
Matrices
• Multiply a matrix by a scalar.
• A scalar simply multiplies all the
terms of a matrix
•
Multiply the following matrices by the given
scalar.
5
•3
6
2 4
•7 8 5
1 7
−1
8
5
• −6 7
−1
4
0
5
5
8
• 5 −2
4
1
5
6
2
Matrices
Add and subtract matrices.
In order to add or subtract matrices the two matrices that you are
adding or subtracting have to have the same dimensions.
Multiply Matrices
• I want a matrix that gives me the cost of medical supplies for each
grade. What are its dimensions? How do I get the matrix?
• Can the matrices with the
following dimensions be
multiplied. If so what is the
dimension of the new matrix
• Draw matrices Below to help you
know. Remember that you are
multiplying rows by columns.
• Can you come up with a rule for
dimensions when you are
multiplying two matrices?
• (3x4)X (4x2)
• (2x1) x (2x2)
• (3x2)x (3x3)
• (1x4)x (4x3)
• (3x1)x (1x3)
• Example of how to
multiply 2X2 matrices.
• Multiply the rows in
the first matrix by the
columns in the second
matrix.
• Practice with multiplying two 2x2 matrices.
−2
•
7
3
6
×
5
4
1
2
• Example of (2X2) x (2x2)
4 2
2 5
•
×
4 5
1 −3
24 14
2𝑥4 + 5𝑥4 2𝑥2 + 5𝑥2
•
=
−8 −13
1𝑥4 − 3𝑥4 1𝑥2 − 3𝑥5
Could you multiply the following matrix.
2 3
• Determine the dimensions of
1 3
•
X
4 1
both matrices
1 4
0 7
• (3 x2) x (2 x3)
• The columns of the first must
2𝑥1 + 3𝑥1
match the rows of the second.
• 4𝑥1 + 1𝑥1
• The Rows of the first and the
columns of the 2nd will be the
5 18 25
dimensions of your new matrix.
• 4 15 15
7
28 49
2
=
7
2𝑥3 + 3𝑥4 2𝑥2 + 3𝑥7
4𝑥3 + 1𝑥3 4𝑥2 + 1𝑥7
Intro to matrices video
• http://ed.ted.com/lessons/howto-organize-add-and-multiplymatrices-bill-shillito
Matrices. Understand that multiplication of
matrices is not commutative
The commutative property states
a(b) will give the same result as
b(a)
Does the commutative property
work for multiplying matrixes.
3 4
−2 2
•
×
1 0
3 1
−2 2
3
•
×
3 1
1
4
0
−4 −8
6 10
•
≠
10 12
−2 2
Matrices associative property works.
Distibutive property for matrices works
Determinant
•
𝑎
𝑐
𝑏
𝑑
a(d) –c(b)
• Practice
11 5
• 1.
3 −2
• 2.
−4 3
1 4
8 2
• 3.
−3 0
The determinant is a number associated with
a square matrix.
• For a 2x2 matrix the
determinant is found by doing
the following
𝑎 𝑏
•
a(d) – c(b)
𝑐 𝑑
Kramer’s rule
• https://www.youtube.com/watc
h?v=jxDgpgdcThU
• The determinant comes from
Kramer’s rule which is just using
elimination to find x and y.
Solve the following equation using Kramers
rule.
• 2x+y = 7
• 3x-2y=7
𝐷𝑒𝑡 𝐴 =
𝑒
𝑓
𝑎
𝑐
2 1
= -4 – 3 = -7
3 −2
𝑏
1
=
𝑑 −2
𝑒 2
𝑓= 3
7
=-7 - 14= -21
7
7
= 14 − 21 = −7
7
Use the Kramer’s rule to solve the following
systems.
Use the Kramer’s rule to solve the following
systems.
• Kayla's school is selling tickets to
the annual dance competition.
On the first day of ticket sales
the school sold 3 senior citizen
tickets and 5 child tickets for a
total of $70. The school took in
$216 on the second day by
selling 12 senior citizen tickets
and 12 child tickets. Find the
price of a senior citizen ticket
and the price of a child ticket.
• The local amusement park is a
popular field trip destination. This
year the senior class at High School
A and the senior class at High
School B both planned trips there.
The senior class at High School A
rented and filled 16 vans and 8
buses with 752 students. High
School B rented and filled 5 vans
and 5 buseswith 380 students.
Each van and each bus carried the
same number of students. How
many students can a van carry?
How many students can a bus
carry?
I can solve a system of equations using
• Set up a system of equations
Matrices.
• Write the equations in
• A theater has 100 seats. The
augmented matrix form.
cost for an adult ticket is 6
dollars and the cost for a student • Use three rules to transform the
matrix
ticket is 4 dollars. If the theater
• A. exchange any two rows
made 460 dollars in one
• B. Multiply a row by any number.
showing, How many adults and
• C. Add any two rows and replace
students were there?
a row with the sum of the two
Your transformed matrix should
1
0
look like this
0 1
• Augmented matrix for equations
• Consider the following system of
equations.
• 3x+4y =12
• 5x-2y =10
• This can be written as an
augmented matrix
3 4 12
•
5 −2 10
Matrices additive identity
• If I had a 2x2 matrix, would 2x2
matrix could I add to my first
matrix to get the same thing?
4 6
•
+
−2 1
4 6
=
−2 1
Matrices Multiplicative identity
• If I had a 2x2 matrix, what 2x2
matrix could I multiply it by to
get the same thing?
2 4
•
x
1 5
2
=
1
4
5
What is an inverse for the following
properties.
• Addition
• Multiplication
• Squaring a number
Matrices Inverses
• What is a matrix inverse
• If A is a matrix then A-1 is the
inverse of A. It undoes A.
• Knowing the inverse will help us
solve matrix equations.
• Find the determinant.
• Find the adjugate.
• Multiply the reciprocal of the
determinant by the adjugate
• This is your inverse equation.
Adjugate
𝑎 𝑏
•𝐴=
𝑐 𝑑
• The adjugate of matrix A is
𝑑 −𝑏
−𝑐 𝑎
3
• 1.
1
7
4
5
• 2.
4
−2
−3
• Explain to your partner how to
get the adjugate.
3.
−3
−2
0
5
Find the inverse of a 2x2 matrix
• Example
• What would happen if we
multiplied A-1 by A?
3 4
•
2 −1
• Determinant = 3(-1) – 2(4)= -11
−1 −4
• Adjugate =
−2 3
• Inverse
1
−11
−1 −4
=
−2 3
1
11
2
11
4
11
−3
11
•
1
11
2
11
4
11
−3
11
x
3 4
=
2 −1
Solve a system using matrices with inverse
matrices.
• A system of equations contains just
as many variables as it does
equations. This means you would
need a square matrix to solve.
• 3 variables would be 3x3 matrix
• 2 variables would be a 2x2 matrix
Solve the system using matrix inverse.
1
11
2
11
• 3x + 4y=1
• 2x –y = -3
•
1
3 4 𝑥
•
x𝑦 =
−3
2 −1
𝑥
1
-1
1
-1
• A xA x 𝑦 =A x
−3
𝑥
• 𝑦 =
4
11
−3
11
3 4 𝑥
x
x𝑦=
2 −1
1
11
2
11
4
11
−3
11
1
x
−3
Practice solve the following system using
inverse matrices
• 2x-y = 2
x + 2y=10
• 1. Write the equation in matrix
form
• 2. Determine the inverse.
• A. determinant
• B. adjugate
• C. Reciprocal of determinant x
adjugate.
• 3. Multiply the inverse by both
sides of the equation.
• 4. Determine the solution
Practice solve the following system using
inverse matrices
• 2x+3y=-1
-3x+y=7
• 1. Write the equation in matrix
form
• 2. Determine the inverse.
• A. determinant
• B. adjugate
• C. Reciprocal of determinant x
adjugate.
• 3. Multiply the inverse by both
sides of the equation.
• 4. Determine the solution
What is the value of using inverse operations
over elimination or substitution.
• Consider the following
systems. What is the same
about them. What is
different about them?
• 2x+3y=8
• 4x-1y = 2
2x+3y=-3
4x – 1y = 9
Another way to solve equations using
matrices (reduced row form)
• We need to
convert our
system to an
augmented
matrix.
Example of row operations.
Start with a 0
2R2 + R1 →R2
2
0
1
7
15
7
You need 1 on
the bottom.
R2 x 1/7
2
0
1
1
15
1
You need a 0 on
the top
R2 x-1+R1→ R1
2
0
0
1
14
1
You need a 1 on
top.
R1 X (.5) → R1
1 0
0 1
7
1
Why use row operations?
• Can you get from the problem to
the identity matrix using row
operations?
You can find the inverse of a matrix with row
operations.
• Just set your square matrix next
to the identity matrix as an
augmented matrix.
• Then use row operations to turn
your square matrix into the
identity matrix.
• Observe what happens to the
identity matrix.
Matrix
Transformation
2 5 1 0
−2 3 0 1
2 5 1 0
0 8 1
−4 0 −2 10
0 2 0
2
−4 0 −2 10
0 1 0
1
1 0 .5 2.5
0 1 0 1
R2 +R1→ R2
5R2-2R1→R1
R2 x.5→ R2
R1 x -.25→R1
Vector
• https://www.youtube.com/watc
h?v=ajTeseYcimE
Vectors show magnitude and direction.
• A vector shows magnitude and
direction.
• A vector can be translated
anywhere on a coordinate plane.
• A vector’s magnitude is measured
by its components.
• Vectors are used in physics to
measure, distance, displacement,
speed, velocity, acceleration, force,
mass, momentum, energy, work,
power.
You can convert a vector into a column matrix
• How did I get the given column
matrix from the vector on the
coordinate plane?
Practice giving a column matrix to each of the
vectors on the coordinate plane.
You can make vector transformations by
multiplying the vector matrix by another matrix.
You can add vectors
• 𝑢+𝑣 =𝑤
• Think of