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Wednesday, April 19, 2023
EQ: What are the similarities and differences between matrices and real numbers?
Warm UpEvaluate each expression for a = -5, b= 1.3,
and c= -7.
1. a + b
2. b – c
3. a – b + c
4. -4b
Solve each equation.
5. 16 = 2x – 5
6. 3y + 8 = -y - 15
-3.7
8.3
-13.3
-5.2
X= 10.5
Y = -5.75
Wednesday, April 19, 2023
EQ: What are the similarities and differences between matrices and real numbers?
Unit 4 Matrices
Lesson 1 Matrix Operations
Essential Question
What are the similarities and differences between matrices and real numbers?
Standards M.ALGII.2.4
Objectives
Vocabulary
Wednesday, April 19, 2023
EQ: What are the similarities and differences between matrices and real numbers?
What we’re learning today:
•What matrices are
•How to add, subtract and multiply matrices.
Unit 4: Lesson 1
Describe how you find your seat in a stadium when you go to a sports game or a concert.
Wednesday, April 19, 2023
EQ: What are the similarities and differences between matrices and real numbers?
The Matrix
Wednesday, April 19, 2023
EQ: What are the similarities and differences between matrices and real numbers?
The Matrix
mnnn
m
m
aaa
aaa
aaa
,2,1,
,22,21,2
,12,11,1
Wednesday, April 19, 2023
EQ: What are the similarities and differences between matrices and real numbers?
Matrix
(Plural: Matrices)
A matrix is a rectangular array of numbers. Matrices are named using capital letters.
Example:
154
702A
Wednesday, April 19, 2023
EQ: What are the similarities and differences between matrices and real numbers?
The Matrix
mnnn
m
m
aaa
aaa
aaa
,2,1,
,22,21,2
,12,11,1
Row 1
Row 2
Row n
Wednesday, April 19, 2023
EQ: What are the similarities and differences between matrices and real numbers?
The Matrix
mnnn
m
m
aaa
aaa
aaa
,2,1,
,22,21,2
,12,11,1
Column 1 Column 2 Column m
Wednesday, April 19, 2023
EQ: What are the similarities and differences between matrices and real numbers?
Dimensions of a Matrix
The dimensions of a matrix are the number of rows and the number of columns in the matrix.
Examples:
154
70223
12
4
3
241
Wednesday, April 19, 2023
EQ: What are the similarities and differences between matrices and real numbers?
The Matrix
mnnn
m
m
aaa
aaa
aaa
,2,1,
,22,21,2
,12,11,1
Entry
(Element)
Wednesday, April 19, 2023
EQ: What are the similarities and differences between matrices and real numbers?
Matrix Entries
AKA: Matrix Elements
The numbers inside a matrix are called entries or elements. The location or “address” of an entry is the number of the row and column where the entry is located.
Example:
The entry at row 2, column 3 is 1
154
702
Wednesday, April 19, 2023
EQ: What are the similarities and differences between matrices and real numbers?
Enter each matrix into your calculator
154
702A
189
315B
01
35C
29
74D
Wednesday, April 19, 2023
EQ: What are the similarities and differences between matrices and real numbers?
Matrix Equality
Two matrices are equal if and only if they have identical dimensions and all corresponding entries are equal.
Example:
154
702
154
702
Wednesday, April 19, 2023
EQ: What are the similarities and differences between matrices and real numbers?
Matrix Addition and Subtraction
It is only possible to add or subtract two matrices, if they have identical dimensions. To find the sum, add corresponding entries. To find the difference, subtract corresponding entries.
Examples:
1105
886
059
184
154
702
72
06
3
4Not Possible
Wednesday, April 19, 2023
EQ: What are the similarities and differences between matrices and real numbers?
Find the difference without a calculator:
29
74
01
35
Wednesday, April 19, 2023
EQ: What are the similarities and differences between matrices and real numbers?
Use your calculator to find each sum or difference:
154
702A
189
315B
01
35C
29
74D
1) A + B2) B + A3) (A + B) + A
4) A + (B + A)
Wednesday, April 19, 2023
EQ: What are the similarities and differences between matrices and real numbers?
Properties of Matrix Addition
For matrices A, B and C with identical dimensions:
Commutative: A + B = B + A
Associative: (A + B) + C = A + (B + C)
Additive Identity: A + 0 = A
Additive Inverse: A + -A = 0
Wednesday, April 19, 2023
EQ: What are the similarities and differences between matrices and real numbers?
The Zero Matrix
A matrix with dimension mn whose entries are all 0 is the additive identity for mn matrices.
Examples:nm,0
00
000 2,2
0000 3,1
Wednesday, April 19, 2023
EQ: What are the similarities and differences between matrices and real numbers?
Scalar Multiplication
A scalar is a real number. To multiply a scalar by a matrix, multiply the scalar by every entry in the matrix.
Example:
31512
2106
154
7023
Wednesday, April 19, 2023
EQ: What are the similarities and differences between matrices and real numbers?
Use your calculator to find each product:
154
702A
189
315B
01
35C
29
74D
1) 5·B2) -3·C3) ½·D
Wednesday, April 19, 2023
EQ: What are the similarities and differences between matrices and real numbers?
Matrix Multiplication
It is only possible to multiply two matrices when the number of columns in the first matrix is equal to the number of rows in the second matrix.
Example:
31512
2106
154
702Not Possible
31512
2106
27
13Possible
23 · 23
22 · 23
Wednesday, April 19, 2023
EQ: What are the similarities and differences between matrices and real numbers?
Matrix Multiplication
The dimensions of the product of two matrices will be the number of rows in the first matrix and the number of columns in the second matrix.
Example:
1533018
66156
31512
2106
27
13
22 · 23 = 23
Wednesday, April 19, 2023
EQ: What are the similarities and differences between matrices and real numbers?
Matrix Multiplication
31512
2106
27
13
12163
Row 1
Column 1
Wednesday, April 19, 2023
EQ: What are the similarities and differences between matrices and real numbers?
Matrix Multiplication
31512
2106
27
13
1218
Row 1
Column 1
Wednesday, April 19, 2023
EQ: What are the similarities and differences between matrices and real numbers?
Matrix Multiplication
31512
2106
27
13
6Row 1
Column 1
Wednesday, April 19, 2023
EQ: What are the similarities and differences between matrices and real numbers?
Matrix Multiplication
31512
2106
27
13
151036
Row 1
Column 2
Wednesday, April 19, 2023
EQ: What are the similarities and differences between matrices and real numbers?
Matrix Multiplication
31512
2106
27
13
1506
Row 1
Column 2
Wednesday, April 19, 2023
EQ: What are the similarities and differences between matrices and real numbers?
Matrix Multiplication
31512
2106
27
13
156
Row 1
Column 2
Wednesday, April 19, 2023
EQ: What are the similarities and differences between matrices and real numbers?
Matrix Multiplication
31512
2106
27
13
31213156
Row 1
Column 3
Wednesday, April 19, 2023
EQ: What are the similarities and differences between matrices and real numbers?
Matrix Multiplication
31512
2106
27
13
363156
Row 1
Column 3
Wednesday, April 19, 2023
EQ: What are the similarities and differences between matrices and real numbers?
Matrix Multiplication
31512
2106
27
13
66156
Row 1
Column 3
Wednesday, April 19, 2023
EQ: What are the similarities and differences between matrices and real numbers?
Matrix Multiplication
31512
2106
27
13
12267
66156
Row 2
Column 1
Wednesday, April 19, 2023
EQ: What are the similarities and differences between matrices and real numbers?
Matrix Multiplication
31512
2106
27
13
2442
66156
Row 2
Column 1
Wednesday, April 19, 2023
EQ: What are the similarities and differences between matrices and real numbers?
Matrix Multiplication
31512
2106
27
13
18
66156
Row 2
Column 1
Wednesday, April 19, 2023
EQ: What are the similarities and differences between matrices and real numbers?
Matrix Multiplication
31512
2106
27
13
1520718
66156
Row 2
Column 2
Wednesday, April 19, 2023
EQ: What are the similarities and differences between matrices and real numbers?
Matrix Multiplication
31512
2106
27
13
30018
66156
Row 2
Column 2
Wednesday, April 19, 2023
EQ: What are the similarities and differences between matrices and real numbers?
Matrix Multiplication
31512
2106
27
13
3018
66156
Row 2
Column 2
Wednesday, April 19, 2023
EQ: What are the similarities and differences between matrices and real numbers?
Matrix Multiplication
31512
2106
27
13
322173018
66156
Row 2
Column 3
Wednesday, April 19, 2023
EQ: What are the similarities and differences between matrices and real numbers?
Matrix Multiplication
31512
2106
27
13
61473018
66156
Row 2
Column 3
Wednesday, April 19, 2023
EQ: What are the similarities and differences between matrices and real numbers?
Matrix Multiplication
31512
2106
27
13
1533018
66156
Row 2
Column 3
Wednesday, April 19, 2023
EQ: What are the similarities and differences between matrices and real numbers?
Matrix Multiplication
31512
2106
27
13
1533018
66156
Wednesday, April 19, 2023
EQ: What are the similarities and differences between matrices and real numbers?
32
15
31
42
Wednesday, April 19, 2023
EQ: What are the similarities and differences between matrices and real numbers?
Use your calculator to find each product:
154
702A
189
315B
01
35C
29
74D
1) C·D
3) D·A4) A·C5) C·A
2) D·C
Wednesday, April 19, 2023
EQ: What are the similarities and differences between matrices and real numbers?
•Worksheet: Multiplying Matrices #1
Wednesday, April 19, 2023
EQ: What are the similarities and differences between matrices and real numbers?
Worksheet: Multiplying Matrices #2