Transcript Matrices

Matrices
Addition and Subtraction
Competency Goal and Objective
• Competency Goal 1: The learner will use
matrices and graphs to model relationships
and solve problems.
• Objective 1.01: Use matrices to model and
solve problems.
• Display and interpret data.
• Write and evaluate matrix expressions to solve
problems.
Matrices
• If you have ever used a spreadsheet
program on the computer, you have worked
with matrices.
• A matrix is a rectangular arrangement of
numbers in rows and columns.
• It is usually described by its dimensions, or
the number of rows and columns, with the
number of rows stated first.
Matrices (cont’d)
• Remember:
• A row travels from left
to right
• A column travels from
top to bottom.
• Each entry in a matrix
is called an element,
which is assigned a
row number and a
column number.
• You try!
Identify the position of
the circled element in
each matrix.
a.
b.
11 15 24
4
L
M
0
M
M
N3
2
1
6
O
P
P
P
Q
Making Matrices
• Say we are planning a pizza and video
party for a few of our friends and we
have to make some decisions about
ordering food. We start by calling the
pizza houses that deliver in the
neighborhood and ask about the price
for pizzas, drinks and salads.
Problem (cont’d)
We could take the information we receive and
record it in a table, like this:
Vin’s
Toni’s
Sal’s
Pizza
$10.10 $10.86 $10.65
Drinks
$1.09
$0.89
$1.05
Salads
$3.69
$3.89
$3.85
Problem (cont’d)
Or, we could write it in matrix (or ordered array) form,
which simply means writing the numbers in a
rectangular array and enclosing them in brackets.
$10.10
L
M
$1.
09
M
M
N$3.69
O
P
P
P
Q
$10.86 $10.65
$0.89 $1.05
$3.89 $3.85
Square matrices
• A matrix, such as the one we have, is known
as a square matrix when the number of rows
(m) equals the number of columns (n).
• Our matrix is a 3 x 3 matrix.
Column and Row Matrices
• If we decide to list
only the prices of Sal’s
offerings, we would
have a column matrix
of dimensions 3 x 1.
$10.65O
L
M
P
$1.
05
M
P
M
N$3.85 P
Q
• When we choose to
look at the pizza prices
alone, they can be
represented with a 1 x
3 row matrix.
$10.10 $10.86 $10.65
Matrix Addition
• Matrix addition is fairly simple! All you
have to do is add entry by entry.
• So, if you had to add these two matrices,
0 1 2OL
6 5 4O
L
M
M
P
P
9 8 7QN
3 4 5Q
N
Adding Matrices
LM0  6
N9  3
O
P
7  5Q
1 5 2  4
84
6 6 6O
L
M
P
N12 12 12Q
Rules for Addition of Matrices
• Remember, that to add matrices they must
have the same number of rows and
columns!
• So a 2 x 3 matrix can not be added to a 2 x 2
matrix.
Subtraction of Matrices
• Subtraction works like addition. It is also
entry-wise. So when given:
1
L
AM
N0
Find:
1. A – B
O
6P
Q
2 0
3
0 4 3 O
L
BM
9 4 3P
N
Q
2. B – A
0 4O
L
CM P
9 4Q
N
3. B - C
Matrix Subtraction
1  0
L
A B  M
N0  9
1
L
M
9
N
2  4
3  4
O
P
9Q
6 3
7
O
P
6  3Q
03
Matrix Subtraction (cont’d)
0 1
L
B A M
90
N
1
L
M
N9
4  2
4  3
6
7
O
P
9 Q
3
O
P
3  6Q
3 0
• Notice, that A-B and B-A are not the same.
1
L
M
9
N
1
O
L
M
P
9 QN
9
6 3
6
7
7
• Subtraction is not commutative!
O
P
9 Q
3
0 4
L
BC  M
9 4
N
0 4 O
O
L
M P
P
3QN
9 4 Q
3
• Not possible, since the dimensions are not
the same.
Your Turn
• Try to solve this problem! Find the values
of x and y.
3 x OL4 6OL1 7 O
L
M P
M P
M
P
2 y 0 QN
3 1QN
5 1 Q
N
• Did you find that
x + 6 =7
x=1
and
2y –3 = -5
y = -1
Problems to Try
1. A trendy garment company receives orders from
three boutiques. The first boutique orders 25
jackets, 75 shirts and 75 pairs of pants. The
second boutique orders 30 jackets, 50 shirts and
50 pairs of pants. The third boutique orders 20
jackets, 40 shirts and 35 pairs of pants. Display
this information in a matrix whose rows
represent the boutiques and whose columns
represent the type of garment ordered. Label the
rows and columns of your matrix accordingly.
Problems (cont’d)
2. For breakfast Patty had cereal, a medium-sized
banana, a cup of 2% fat milk and a slice of
buttered toast. She recorded the following
information in her food journal. Cereal: 165
calories, 3 g fat, 33 g carbohydrate, and no
cholesterol. Banana: 120 calories, no fat, 26 g
carbohydrate and no cholesterol. Milk: 120
calories, 5 g of fat, 11 g carbohydrate, and 15
mg cholesterol. Buttered toast: 125 calories, 6 g
fat, 14 g carbohydrate and 18 mg cholesterol.
Display this information as a matrix.
Answers
2. Answers may vary
1. Matrix
Jackets
1st Boutique
2nd Boutique
3rd Boutique
L
M
M
M
20
N
Shirts
Pants
O
P
P
35P
Q
25 75 75
30 50 50
40
165
L
M
3
M
M
33
M
N0
165
L
M
120
M
M
120
M
125
N
O
6 P
P
14 P
P
18 Q
120 120 125
0
5
26
11
0
15
3 33
0 26
5 11
6 14
O
0P
P
15P
P
18Q
0