Transcript In the context of transformations

World Cup
Matrix Multiplication….
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Below is a league table for the group stage of the World Cup 2010.
The top 2 teams in each group progress through to the next round;
scoring 3 points for a win, 1 point for a ‘draw’ and 0 points for a
losing a match
Matrix multiplication can be used 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 write down the matrix that represents
the points awarded.
w d
U 1

E 1
S 1

A0

l
2 0
5
 3  
2 0  5
1 

1 1   4
0  
1 2
1
Its important to make sure that the correct points line
up with the appropriate column.
Matrix multiplication
This ‘structure’ can be helpful:
 2 3 1 


 1 2  4
 1 4


 2 1 
Inverse of a 2 x 2 matrix
When a matrix is multiplied by its inverse, the answer is the identity
-1
AA
=I
a b 
A

c d 
-1
=A A
? ?
A 

? ?
1
-1
AA
=I
-1
=A A
Here are some examples – can you find the pattern?
 2 3   2  3
1 0
1 2 
0 1
=


  1 2 


1  2   3 2 = 1 0
 2  3   2 1
0 1





1 4   5  4  =  1 0 
1 5    1 1   0 1 

 
 

 2 4
1 3


  2 6
 1 1 


 2  3
2 2 


The determinant and the Inverse of a 2 x 2 matrix
The determinant of A is ad - bc
a b 
A

c d 
The inverse of A is:
1  d b 
A 


ad  bc  c a 
-1
e.g. Find the inverse of this matrix
 3 2 
A

1 2 
1  2 2
A  

8  1 3 
-1
Inverse of a 2 x 2 matrix
What is the determinant
of this matrix?
Comments?
singular
 4 2 


6

3


In the context of transformations …
y
4
2
x
–2
2
4
6
Use the transformations represented by the matrices below to find
the image of the triangle above:
 1  2


 2 4 
 2  6


 1  3
What happens in each case?
 2 4


 3 6
In the context of transformations …
y
6
Eve ry point on the obje ct ( a nd e ve ry othe r point in the pla ne ) is ma ppe d on to the line y = - 0 .5 x.
4
2
–5
 2  4


1  2
5
–2
–4
has zero determinant
x
10
15
In the context of transformations…