Transcript Basic Operations With Matrices Notes Week 1x

Unit 1 Objectives
Matrix: a rectangular array of numbers or variables used to
organize data.
 2 4 
 3 5 


1 6


The numbers in the matrix are called
elements. There are 6 elements in this
matrix.
Matrix dimensions: The dimensions tell how many
ROWS & COLUMNS there are in the matrix.
• Dimensions
• A 3x 2
• REMEMBER:
• “RC cola”
• Rows by columns
 2 4 
 3 5 


1 6


Organizing Data Into Matrices
• Writing the Dimensions of a Matrix
a) 2
-3
18
This matrix has two rows and two
7
columns, therefore it is a 2 x 2 matrix
(read as two-by-two)
0
6 -10
three
a1x3
one by three)
b)
This matrix has one row and
columns, therefore it is
matrix (read as
Matrix Position
We can give the location of an element in a matrix
by naming its row, then its column.
 2 4 8 4 
 3 5 6 3 


 1 6 2 0 


3 is in position __𝐴24 __
6 is in position ____
4 is in position ____
-2 is in position _____
Organizing Data Into Matrices
Identify each matrix element.
K =
3 –1 –8 5
1 8 4 9
8 –4 7 –5
a. k12
a. K =
b. k32
3 –1 –8 5
1 8 4 9
8 –4 7 –5
c. k23
b. K =
d. k34
3 –1 –8 5
1 8 4 9
8 –4 7 –5
k12 is the element in the first row
and second column.
k32 is the element in the third row
and second column.
Element k12 is –1.
Element k32 is –4.
4-1
Special Matrices
Row Matrix: has only 1 row.
1
Column Matrix: has only 1 column.
2
-6
2
 6 
 
3
 
5
Square Matrix: Has the same number of rows and
columns.
 8 6 5 
 2 3 7 


1 0 9 


Adding and Subtracting Matrices
The table shows information on ticket sales for a new
movie that is showing at two theaters. Sales are for children (C)
and adults (A).
Matinee
Theater C
1
198
2
201
A
350
375
Evening
C
54
58
A
439
386
a. Write two 2  2 matrices to represent matinee and evening sales.
Theater 1
Theater 2
Matinee
C
A
198 350
201 375
Theater 1
Theater 2
4-2
Evening
C
A
54 439
58 386
Adding and Subtracting Matrices
(continued)
b. Find the combined sales for the two showings.
198
201
=
350
375
Theater 1
Theater 2
+
54
58
439
386
C
252
259
A
789
761
=
4-2
198 + 54
201 + 58
350 + 439
375 + 386
Warm-UP
A= 7
9
C=
7
9
-8
-3
6
-4
7
8
10
-5
B=
-1
7
0
9
-3
-6
Find: A22 and C31
Find: the dimensions of each
Matrix
Find: A + B and B – A and C + B
Practice in Pairs
Transpose Matrix: A matrix which is formed by
turning all the rows of a given matrix
into columns and vice-versa. The transpose of
matrix A is written AT.
1 4
1 2 3 T is
2 5
4 5 6
3 6
Equal Matrices: Two matrices are equal if they have
the same dimensions and the corresponding
elements are equal.
Multiplying a Matrix by a Scalar
Scalar Multiplication of Matrices
 2 8   4( 2 ) 4( 8 )   8 32 
4





 5 7   4( 5 ) 4( 7 )   20 28 
 15 12 10   3( 15 ) 3( 12 ) 3( 10 )   45 36 30 
3  10 20 0    3( 10 ) 3( 20 ) 3( 0 )    30 60 0 
 5
2 6   3( 5 )
3( 2 ) 3( 6 )   15 6 18 

Practice
7
-10
8
9
8
0
-3
-6
11
-4
=
Adding and Subtracting Matrices
Example
5
4
-7
10
3
+
7
-2
4
0
11
11
2
=
Example
Using Adding and Subtracting Matrices in Equations
ALGEBRA 2 LESSON 4-2
Solve X –
X –
X –
2 5
3 –1
8 0
+
2 5
3 –1
8 0
2 5
3 –1
8 0
2 5
3 –1
8 0
=
=
10 –3
–4 9
6 –9
=
10 –3
–4 9
6 –9
10 –3
–4 9
6 –9
+
for the matrix X.
2 5
3 –1
8 0
Add
2 5
3 –1
8 0
to each side of
the equation.
X =
12 2
–1 8
14 –9
4-2
Simplify.
Worksheet
Warm-Up
Solve
A=
2
4
0
7
B= 4 -1 2 Find: A - B
1 6 -3
C-A
A-C
4B
C=
D= 2
12
-1
E= 4
3
1
3
Adding and Subtracting Matrices
ALGEBRA 2 LESSON 4-2
Solve the equation
2m – n
–3
8
–4m + 2n
=
15
8
m+ n
–30
2m – n
–3
8
–4m + 2n
=
15
8
m+ n
–30
2m – n = 15
–3 = m + n
for m and n.
–4m + 2n = –30
Since the two matrices are equal, their corresponding elements are equal.
4-2
Adding and Subtracting Matrices
ALGEBRA 2 LESSON 4-2
(continued)
Solve for m and n.
2m – n = 15
m + n = –3
3m = 12
m = 4
4 + n = –3
n = –7
Add the equations.
Solve for m.
Substitute 4 for m.
Solve for n.
The solutions are m = 4 and n = –7.
Transparencies
4-2
Challenge Problem
Multiplying Matrices
We can only multiply matrices if the number of
COLUMNS in the FIRST matrix is the equal to the number
of ROWS in the SECOND matrix.
Multiplying Matrices
• Step 1 : Multiply the elements in the first row of A with the
corresponding elements in the first column of B. Add the
products to get the element C 11
• Step 2 : Multiply the elements in the first row of A with the
corresponding elements in the second column of B. Add
the products to get the element C 12
Multiplying Matrices continued
• Step 3 : Multiply the elements in the second row of A with
the corresponding elements in the first column of B. Add
the products to get the element C 21
• Step 4 : Multiply the elements in the second row of A with
the corresponding elements in the second column of B.
Add the products to get the element C 22
Example
How do we multiply matrices? What is A x B?
1
A
2
2
0
-1
1
3
x
1
B
0
-1
-1
3
Silent Appointment #1
10 6 


 5 2 1  1 0
 3 9 
Silent Appointment #2
Warm-Up