Matrix Intro
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Transcript Matrix Intro
Algebra 2 – matrices intro
Ms. Riling’s matrix
FOOD TYPE
BEEF
OTHER
MEAT
DAIRY
FRUITS AND BREADS,
VEGGIES
COOKIES
BREAKFAST
0
0
0
5
8
LUNCH
0
0
0
5
7
DINNER
0
0
0
2
9
DESSERT
0
0
0
5
2
OTHER
0
0
2
12
6
W’s matrix
FOOD TYPE
BEEF
OTHER
MEAT
DAIRY
FRUITS AND BREADS,
VEGGIES
COOKIES
BREAKFAST
0
4
6
5
3
LUNCH
5
0
3
5
5
DINNER
4
0
0
2
2
DESSERT
0
0
0
5
2
OTHER
0
0
1
4
3
1. What if there were 10 people who ate JUST like
you?
-Make one matrix representing what you all ate
2. With a partner, figure out how much you ate in
total
-Make one matrix for the data
3. A matrix for the food you and your partner would
eat over 2 days
4. You buy exactly the amount of food you need.
Your partner eats it, following their usual eating
habits.
-Write a matrix for what’s left. (If they need extra, write
how much more it should be, but make it negative to
differentiate)
Matrices
• Definition: a 2D array of data
a rectangular arrangement of
numbers in rows and columns
• Dimensions: An “n x m matrix” has n rows and
3 columns
5 2 11
9
4
5
2 3 4 2 4
2 9 8
8 1 6 7 5
3 1 6
• Equal matrices: matrices with the same
dimensions, and the same entry in every spot
• Nomenclature: we refer to matrices using
capital letters – typically A, B, C
Matrix Elements
• Each number in the matrix is called an ELEMENT
• To identify it:
– Describe which row and column it is in
1 4 7
A 2 5 8
3 6 9
a23 8
a21 __
a33 __
• We always describe rows first, then columns
Adding and subtracting matrices
• Just add or subtract the
numbers in the same positions
• What needs to be true about two
matrices if you want to add
them together?
a b e
c d g
f a e b f
h c g d h
a b e
c d g
f a e b f
h c g d h
• Let’s try some subtraction:
23 9 7.1 4 10 .1
0 3 6 70 3 1
Scalar Multiplication
• How did you create a matrix describing the food 10 people just like you
ate?
• Scalar: a number you multiply the matrix by
* Always write the scalar to the matrix’s left
• Just multiply all the elements by the scalar:
2 5 4 2 4 5 8 20
4
3
1
4
3
4
1
12
4
• Try one:
6
3
2
1
0
1
2
0
1
Rules of Arithmetic
• Associative Property: (A + B) + C = A + (B + C)
• Commutative:
Now, about food…
• How much land did each meal use up?
• We can measure the land that were needed to produce your
food. That depends on the animals involved, the food they
ate, the machines used, etc.
• Since we didn’t measure whether your food was
local/organic/etc, we aren’t going to be very precise.
The units
• We measure land in HECTARES (ha)
• 1 ha = about 2.5 acres
The Data
FOOD
HECTARES USED PER
OUNCE OF FOOD
Beef
.0157
Other meat
.0032
Dairy
.0115 (Note: milk is
actually less)
.0004
Fruits and veggies
Baked things
.0015 (Note: bread is
actually less)
Matrix Multiplication
• We want to multiply each food type by the
amount of ha that food type uses.
• You created a matrix representing the food
you ate:
0 0 8
3 0 0
0 3 3
0 0 3
0 0 2
• Now make onefor
land use:
3
6
9
5
3
3
2
5
0
5
.0157
.0032
.0115
.
0004
.0015
Now we just have to multiply them:
0
3
0
0
0
0
0
3
0
0
8
0
3
3
2
3
6
9
5
3
3
2
5
0
5
.0157
.0032
.0115
.
0004
.0015
.0977
.0525
.0252
.
0365
.0317
What does each row in your new matrix
represent?
Other thoughts?
Multiplying Matrices
a b e
c d g
f ae bg
h ce dg
af bh
cf dh
Say we multiply matrices A and B to get a new matrix, C
To get element cmn, you need to multiply
row m in A
by
column n in B
Practice – multiplying matrices
5
7
3 2 6 11
9
6 4
5
1 8
2
3 2
5
4 3 5 17
3 1 2 7 1
9
1 5
2 1 4
2 2
6 4 3
0 1
More practice
5 61 3
2 34 2
5
1 3
3
4 2
6
1 35 6
4 2 2 3
6 2 2
2 5 13 1 1
4 0 1
What needs to be true about two matrices if you want to
multiply them?
Does the order of the matrices matter?
What size will the solution matrix be?