FP1 matrices lesson 1
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Transcript FP1 matrices lesson 1
Further Pure 1
Lesson 1 – Matrices
Working & Transformations
Definitions
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A matrix is just a rectangle of numbers.
It’s a bit like a two-way table.
You met this concept briefly in D1.
The matrix below shows how many arcs exist
between each node.
Definitions Definitions
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Here are more examples of matrices.
3 1
= 2 × 2
2 5
2
0
3
1
=4×1
2 3 6 2 3
= 2 × 5
4 4 0 1 5
1 0 3
2 9 0 = 3 × 3
5 2 7
You can see that each matrix is a different size.
The size (or order) of a matrix is given as rows × columns.
What is the order of each of the above matrices?
Definitions
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Here are two special types of matri that you need to
be familiar with.
The identity matrix, I.
1 0 0
1 0
I3 0 1 0
I2
0 1
0 0 1
1 0 .. 0
0
1
:
In
:
:
0 .. .. 1
The zero matrix, O.
0 0
O2
0
0
0 0 0
O3 0 0 0
0 0 0
0 0 .. 0
0
0
:
On
:
:
0 .. .. 0
Definitions
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Matrices with the same number of rows and
columns are known as square matrices.
2 2
1 6
Identity matrices are
always square as the
1`s on the diagonal
must run from corner
to corner.
1 0 3
2 9 0
5 2 7
1
0
0
0
0
1
0
0
0
0
1
0
0
0
0
1
Definitions
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Two matrices are equal if:
They have the same order
Each element in one matrix is equal to the
corresponding element in the other matrix.
Using Matrices (+/-)
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We can add and subtract matrices only if they have
the same order.
All you do is add or subtract corresponding elements.
2 2 1 2 0 0 4 2 1
3 0 5 7 1 8 10 1 3
1
2 2 1 2 0 0 0 2
3 0 5 7 1 8 4 1 13
Why can you not add matrices with different orders?
Using Matrices (×)
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You can multiply a matrix by a number as
illustrated in the example below.
3
2 2 1 6 6
3
3 0 5 9 0 15
All that has happened is each element has been
multiplied by the number outside the matrix.
In general for any 2 × 2 matrix.
p q ap aq
a
r s ar as
Remember that this will work for any matrix of any
order.
Problems
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Explain why matrix addition is
Commutative, i.e. A + B = B + A
Associative, i.e. A + (B + C) = (A + B) + C
Addition in elements is both commutative and
associative.
Do Ex 1A pg 3
Using Matrices (×)
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Sometimes you can multiply two matrices together.
However not all matrices can be multiplied together.
Lets imagine 2 matrices called A and B.
If we want to calculate A × B then A must have the
same number of columns as B has rows.
The sum you do is multiply each element in the 1st
row of A by each element in the first column of B,
then add together your answers.
You then do the same for all the row and column
combinations.
On the next slide is a worked example.
Using Matrices (×)
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Let A be a 2 × 3 matrix and B be a 3 × 2 matrix.
a1 a2 a3
A
a 4 a5 a 6
So A × B is given by
a1 a2
a 4 a5
b1 b 4
B b 2 b5
b b
6
3
b1 b 4
a1b1 a2b2 a3b3
a3
b2 b5
a6
a 4b1 a5b2 a6b3
b3 b 6
a1b 4 a2b5 a3b6
a 4b 4 a 5b 5 a 6b 6
Now take every element in the first row of A and multiply them
by every element in the first column of B, adding your answers.
Now repeat with the 2nd row of A and 1st column of B.
Next 1st row of A and 2nd column of B.
Finally 2nd row of A and 2nd column of B.
Using Matrices (×)
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Try the numerical example below.
2 3 1
A
1 2 4
So A × B is given by.
1 2
B 2
0
4 5
1 2
4
0
2 3 1
2
0
1 2 4 4 5 19 18
Now take every element in the first row of A and multiply them
by every element in the first column of B, adding your answers.
Now repeat with the 2nd of A and 1st column of B.
Next 1st row of A and 2nd column of B.
Finally 2nd row of A and 2nd column of B.
Using Matrices (×)
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What would happen if you found B × A.
2 3 1
A
1 2 4
So B × A is given by.
1 2
B 2
0
4 5
9
1 2
4 1
2 3 1
4
0
6
2
2
4 5 1 2 4 9 2 24
What do you notice about this answer compared to the
last?
From these examples we can conclude that AB = BA
So matrix multiplication is not commutative
Using Matrices (×)
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If two matrices A and B have orders p × q and q × r
respectively then A × B does exist, and will have
order p × r.
p
q ×
q
r
=
p
Note: In this case B × A does not exist.
q
p ×
r
q
r
Associative
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Use the following matrices to show that matrix
multiplication is associative.
i.e. A(BC) = (AB)C
2 1 3 0
A
4 0 4 3
0
3
B
4
6
5 1
2 4 0
9 0
C 2 3 9
0 1
0 5 1
2 6
Using Matrices (×)
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Note that any matrix multiplied by the identity
matrix is itself.
2 1 6 1 0 0 2 1 6
3 1 8 0 1 0 3 1 8
6 9 3 0 0 1 6 9 3
And any matrix multiplied by the zero matrix
is the zero matrix.
2 1 6 0 0 0 0 0 0
3 1 8 0 0 0 0 0 0
6 9 3 0 0 0 0 0 0
Questions
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Try some of the following multiplications.
1 1 4 2 10
2
3
2
0
2
3
8
2 5 8
1 3
4 1 7
1 0
0
0
2 3 6
3
1 0 20
6 1 0
8
2
7
2
3
4
2
1 5 1 5 4 10
6
2
0
3
4
1
3
2
1
1 2 0 0 2 5
29
10
4
31
2
3
Example
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Below is a league table for the group stage of the
world cup 2006.
The top 2 teams in each group progress through to
the next round.
Team
MP
W
D
L
England
3
2
1
0
Paraguay
3
1
0
2
Sweden
3
1
2
0
3
0
1
2
Trinidad and
Tobago
Use matrix multiplication to calculate the final points
and hence state who progressed through to the next
round.
Solution
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We can write the league table as a matrix.
Next we can add the
w d l
matrix that represents
E 2 1 0
7
the points awarded.
3
P 1 0 2 3
Its important to make
1
sure that the correct
S 1 2 0
5
0
points line up with the
1
T 0 1 2
appropriate column.
Now we can multiply the two matrices
together.
This shows us that England and Sweden
progressed through to the next group.
Summary
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Matrix addition is Commutative
Matrix addition is Associative
Matrix multiplication is not Commutative
Matrix multiplication is Associative
AI = A
AO = O