Lesson 25
NYS COMMON CORE MATHEMATICS CURRICULUM
M1
PRECALCULUS AND ADVANCED TOPICS
Lesson 25: Matrix Multiplication and Addition
Classwork
Opening Exercise
4
Consider the point ( ) that undergoes a series of two transformations: a dilation of scale factor 4 followed by a
1
reflection about the horizontal axis.
a.
What matrix produces the dilation of scale factor 4? What is the coordinate of the point after the dilation?
b.
What matrix produces the reflection about the horizontal axis? What is the coordinate of the point after the
reflection?
c.
Could we have produced both the dilation and the reflection using a single matrix? If so, what matrix would
both dilate by a scale factor of 4 and produce a reflection about the horizontal axis? Show that the matrix you
came up with combines these two matrices.
Example 1: Is Matrix Multiplication Commutative?
a.
𝜋
2
Take the point ( ) through the following transformations: a rotation of and a reflection across the 𝑦-axis.
1
2
Lesson 25:
Matrix Multiplication and Addition
This work is derived from Eureka Math ™ and licensed by Great Minds. ©2015 Great Minds. eureka-math.org
This file derived from PreCal-M1-TE-1.3.0-07.2015
S.127
This work is licensed under a
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
Lesson 25
NYS COMMON CORE MATHEMATICS CURRICULUM
M1
PRECALCULUS AND ADVANCED TOPICS
b.
Will the resulting point be the same if the order of the transformations is reversed?
c.
Are transformations commutative?
d.
Let 𝐴 = (
e.
Is matrix multiplication commutative?
f.
2
If we apply matrix 𝐴𝐵 to the point ( ), in what order are the transformations applied?
1
g.
2
If we apply matrix 𝐵𝐴 to the point ( ), in what order are the transformations applied?
1
h.
2
Can we apply ( ) to matrix 𝐵𝐴?
1
0
1
−1
−1
) and 𝐵 = (
0
0
Lesson 25:
0
). Find 𝐴𝐵 and then 𝐵𝐴.
1
Matrix Multiplication and Addition
This work is derived from Eureka Math ™ and licensed by Great Minds. ©2015 Great Minds. eureka-math.org
This file derived from PreCal-M1-TE-1.3.0-07.2015
S.128
This work is licensed under a
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
Lesson 25
NYS COMMON CORE MATHEMATICS CURRICULUM
M1
PRECALCULUS AND ADVANCED TOPICS
Exercises 1–3
1.
2.
a.
1 0
4
) and 𝑀 = (
0 1
3
Find 𝐼𝑀.
b.
Find 𝑀𝐼.
c.
Do these results make sense based on what you know about the matrix (
Let 𝐼 = (
−6
).
−2
1
0
0
)?
1
Calculate 𝐴𝐵 and then 𝐵𝐴. Is matrix multiplication commutative?
2 0
1 5
a. 𝐴 = (
), 𝐵 = (
)
−2 3
0 1
b.
𝐴= (
−10
3
1
−3
),𝐵 = (
7
4
Lesson 25:
2
)
−1
Matrix Multiplication and Addition
This work is derived from Eureka Math ™ and licensed by Great Minds. ©2015 Great Minds. eureka-math.org
This file derived from PreCal-M1-TE-1.3.0-07.2015
S.129
This work is licensed under a
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
NYS COMMON CORE MATHEMATICS CURRICULUM
Lesson 25
M1
PRECALCULUS AND ADVANCED TOPICS
3.
Write a matrix that would perform the following transformations in this order: a rotation of 180°, a dilation by a
2
scale factor of 4, and a reflection across the horizontal axis. Use the point ( ) to illustrate that your matrix is
1
correct.
Example 2: More Operations on Matrices
Find the sum. (
2
−2
0
1
)+(
3
0
Find the difference. (
Find the sum. (
2
−2
2
−2
5
)
1
0
1
)−(
3
0
0
0
)+(
3
0
5
)
1
0
)
0
Exercises 4–5
4.
Express each of the following as a single matrix.
6 −3
−2
8
a. (
)+(
)
10 −1
3 −12
Lesson 25:
Matrix Multiplication and Addition
This work is derived from Eureka Math ™ and licensed by Great Minds. ©2015 Great Minds. eureka-math.org
This file derived from PreCal-M1-TE-1.3.0-07.2015
S.130
This work is licensed under a
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
NYS COMMON CORE MATHEMATICS CURRICULUM
Lesson 25
M1
PRECALCULUS AND ADVANCED TOPICS
5.
b.
(
−2 7 3
1
)( ) + ( )
−3 1 4
5
c.
(
4
8 5
)−(
−3
0 15
−6
)
18
In arithmetic, the additive identity says that for some number 𝑎, 𝑎 + 0 = 0 + 𝑎 = 0. What would be an additive
identity in matrix arithmetic?
Lesson 25:
Matrix Multiplication and Addition
This work is derived from Eureka Math ™ and licensed by Great Minds. ©2015 Great Minds. eureka-math.org
This file derived from PreCal-M1-TE-1.3.0-07.2015
S.131
This work is licensed under a
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
Lesson 25
NYS COMMON CORE MATHEMATICS CURRICULUM
M1
PRECALCULUS AND ADVANCED TOPICS
Lesson Summary
𝑝
𝑎 𝑐
) and 𝑀 is given by (𝑞
𝑏 𝑑
𝑥
𝑝𝑎 + 𝑟𝑏 𝑝𝑐 + 𝑟𝑑
(
) to (𝑦).
𝑞𝑎 + 𝑠𝑏 𝑞𝑐 + 𝑠𝑑
𝑎 𝑐
1
If 𝐿 is given by (
) and 𝐼 is given by (
𝑏 𝑑
0
𝑎 𝑐
0
If 𝐿 is given by (
) and 𝑂 is given by (
𝑏 𝑑
0
𝑂 + 𝐿 = 𝐿 + 𝑂 = 𝐿.

If 𝐿 is given by (


𝑟
𝑥
𝑠 ), then 𝑀𝐿 (𝑦) is the same as applying the matrix
0
), then 𝐼 acts as a multiplicative identity, and 𝐼𝐿 = 𝐿𝐼 = 𝐿.
1
0
), then 𝑂 acts as an additive identity, and
0
Problem Set
1.
What type of transformation is shown in the following examples? What is the resulting matrix?
cos (𝜋) −sin (𝜋) 3
a. (
)( )
sin (𝜋) cos (𝜋) 2
b.
c.
d.
e.
f.
0 −1 3
)( )
1 0
2
3 0 3
(
)( )
0 3 2
−1 0 3
(
)( )
0 1 2
1 0
3
(
)( )
0 −1 2
cos (2𝜋) −sin (2𝜋) 3
(
)( )
sin (2𝜋) cos (2𝜋) 2
(
√2
g.
(2
√2
2
√3
h.
( 2
1
2
−√2
2 ) ( 3)
√2
2
2
− 12
√3
3
)( )
2
2
Lesson 25:
Matrix Multiplication and Addition
This work is derived from Eureka Math ™ and licensed by Great Minds. ©2015 Great Minds. eureka-math.org
This file derived from PreCal-M1-TE-1.3.0-07.2015
S.132
This work is licensed under a
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
NYS COMMON CORE MATHEMATICS CURRICULUM
Lesson 25
M1
PRECALCULUS AND ADVANCED TOPICS
2.
Calculate each of the following products.
2 3 2
a. (
)( )
5 4 3
3 2 3 2
b. (
)(
)
4 5 1 0
−1 −3 −3
c. (
)( )
−2 −4 −1
−3 −1 −2 −1
d. (
)(
)
−4 −2
0 −3
5 0 0
e. (
)( )
−1 2 3
3 −1
1
0
f. (
)(
)
4 −2 −2 −3
3.
Calculate each sum or difference.
1 3
4 2
a. (
)+(
)
2 4
3 1
−2 3
−4 −5
b. (
)+(
)
−1 4
−6 −7
3
−5
c. ( ) + ( )
2
4
3
7
d. ( ) − ( )
9
−5
−2 3
−4 −5
e. (
)−(
)
−1 4
−6 −7
4.
1
In video game programming, Fahad translates a car, whose coordinate is ( ), 2 units up and 4 units to the right,
1
𝜋
𝜋
rotates it radians counterclockwise, reflects it about the 𝑥-axis, reflects it about the 𝑦-axis, rotates it radians
2
2
counterclockwise, and finally translates it 4 units down and 2 units to the left. What point represents the final
location of the car?
Lesson 25:
Matrix Multiplication and Addition
This work is derived from Eureka Math ™ and licensed by Great Minds. ©2015 Great Minds. eureka-math.org
This file derived from PreCal-M1-TE-1.3.0-07.2015
S.133
This work is licensed under a
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.