3.6: Multiply Matrices

3.6: Multiply Matrices
1.
Objectives:
To multiply matrices
Assignment:
• P. 199-202: 1, 2-36
even, 37, 38, 40, 42,
43, 46, 47
• Challenge Problems: 111
Warm-up
1. How do you find the dimensions of a
matrix?
2. We’ve learned how to add and subtract
matrices and how to multiply by a scalar.
How do you think we should multiply
matrices?
Objective 1
You will be able to
multiply matrices
Matrix Multiplication
Multiplying two matrices is
weird. Thank you, Arthur
Cayley (1858).
Because of the way it is
defined, if we are
multiplying two matrices
𝐴 × 𝐵, then the number of
columns in 𝑨 must match
the number of rows in 𝑩.
Arthur Cayley, 1850ish
Matrix Multiplication
3 9
4 2
If 𝐴 = 2 8 and 𝐵 =
,
6 7
1 5
Order: 𝟑 × 𝟐
Must match
Ξ
then 𝐴𝐵 = Ξ
Ξ
Order:
𝟐×𝟐
Ξ
Ξ .
Ξ
𝟑×𝟐
Arthur Cayley, 1850ish
Getting Rid of the Middle Man
In general, if matrix 𝐴 has order 𝒎 × 𝒏 and 𝐵
has order 𝒏 × 𝒑, then 𝐴𝐵 has order 𝒎 × 𝒑.
When multiplying two matrices, the first thing you
should do is find the order of the product!
Exercise 1
Can you find 𝐴𝐵? If so, what is the order of
the product?
1.
𝐴: 5 × 2; 𝐵: 2 × 2
2.
𝐴: 3 × 2; 𝐵: 3 × 2
Exercise 2a
Multiply.
4
2 3
5
Exercise 2b
Multiply.
2
4 5
3
Exercise 2c
Multiply.
2
1 4
3
2 5
Exercise 2d
Multiply.
2 3 1 4
1 4 2 5
Exercise 2d
Multiply.
2∙1+3∙2 2∙4+3∙5
2 3 1 4
=
1∙1+4∙2 1∙4+4∙5
1 4 2 5
2×2
2×2
2×2
=
8
9
23
24
Exercise 2d
Multiply.
2∙1+3∙2 2∙4+3∙5
2 3 1 4
=
1∙1+4∙2 1∙4+4∙5
1 4 2 5
2×2
2×2
2×2
=
8
9
23
24
Exercise 2d
Multiply.
2∙1+3∙2 2∙4+3∙5
2 3 1 4
=
1∙1+4∙2 1∙4+4∙5
1 4 2 5
2×2
2×2
2×2
=
8
9
23
24
Exercise 2d
Multiply.
2∙1+3∙2 2∙4+3∙5
2 3 1 4
=
1∙1+4∙2 1∙4+4∙5
1 4 2 5
2×2
2×2
2×2
=
8
9
23
24
Exercise 2d
Multiply.
2∙1+3∙2 2∙4+3∙5
2 3 1 4
=
1∙1+4∙2 1∙4+4∙5
1 4 2 5
2×2
2×2
2×2
=
8
9
23
24
Matrix Multiplication
Definition of Matrix Multiplication
C
o
l
Row x
u
m
This definition will make more sense
n
once we do an application question.
Matrix Multiplication
Definition of Matrix Multiplication
C
o
RRooww x l = RC + Ro + ol + ou + wm + wn
u
m
n
The Order of the Product
So you might be asking why the number of
columns of the first matrix must equal the
number of rows in the second matrix.
This is because we must multiply a row (in the
first matrix) by a column (in the second matrix),
and the number of elements must match.
How many elements are there in any row in the
first matrix? How many elements are there in
any column in the second matrix?
Exercise 3a
Two softball teams submit equipment lists to
their sponsors.
Bats
Balls
Gloves
Woman’s
Team
12
45
15
Men’s Team
15
38
17
If a bat costs $21, a ball costs $4, and a
glove costs $30, use matrices to find the
total cost of equipment for each team.
Exercise 3b
The previous Exercise simplifies to the following
matrix multiplication problem. Can you now
see why matrix multiplication is so weird?
21
12 45 15
4
15 38 17
30
Exercise 4
𝐴=
1 0
−3 3
1
5
,𝐵=
, and 𝐶 =
1 −2
0 1
−3 −2
1. Find 𝐴𝐵
2. Find 𝐵𝐴
3. Find 𝐶𝐴
Exercise 5
𝐴=
1 0
−3 3
1
5
,𝐵=
, and 𝐶 =
1 −2
0 1
−3 −2
1. Find 𝐴 𝐵 + 𝐶
2. Find 𝐴𝐵 + 𝐴𝐶
Properties of Matrix Multiplication
Let 𝐴, 𝐵, and 𝐶 be matrices and let 𝑘 be a
scalar.
Associative Property of Matrix
Multiplication
Left Distributive Property
Right Distributive Property
Associative Property of Scalar
Multiplication
𝐴 𝐵𝐶 = 𝐴𝐵 𝐶
𝐴 𝐵 + 𝐶 = 𝐴𝐵 + 𝐵𝐶
𝐴 + 𝐵 𝐶 = 𝐴𝐶 + 𝐵𝐶
𝑘 𝐴𝐵 = 𝑘𝐴 𝐵 = 𝐴 𝑘𝐵
Remember: There is no commutative property for matrix
multiplication!
Identity Matrix
Recall that multiplying a number by 1 gives
you back the same number. This is the
Identity Property of Multiplication.
Similarly, multiplying a matrix by the identity
matrix, 𝐼, will return the original matrix.
Identity Property of Matrix
Multiplication
𝐴𝐼 = 𝐼𝐴 = 𝐴
Identity Matrix
The 𝑛 × 𝑛 identity matrix has 1s along the
main diagonal and zeros everywhere else.
Exercise 6
What is the identity matrix for a 3 × 3?
Exercise 7
Find 𝐴2 .
𝐴=
−2 1
−8 4
Exercise 8
Find 𝐴2 .
2
5
𝐴 = −1 4
3 −7
3.6: Multiply Matrices
1.
Objectives:
To multiply matrices
Assignment
• P. 199-202: 1, 236 even, 37, 38,
40, 42, 43, 46, 47
• Challenge
Problems: 1-11
“I’m always hiding the remote control.”