J20

Find the matrix which represents an enlargement of scale factor 2 with centre O
Solution
The images of I and J are I 1 (2, 0) and J 1 (0, 2), respectively. Therefore, the matrix
representing an enlargement of scale factor 2 about O is
2
0
0
2
In general, an enlargement of scale factor k with O as centre of enlargement is
k
0
represented by the matrix
0
k
Note: Finding the matrix of transformation using the above method (based on the
images of points I and J) works only when the transformation is a reflection, rotation,
enlargement, shear or stretch in which the origin remains fixed.
Finding the matrix of transformation by calculation.
Example
Triangle ABC with vertices A(2, 1), B(2, 3) and C(3, 1) maps onto A 1(4, 1),
B 1(8, 3) and C 1(5, 1) under a shear with the x-axis as the invariant line and shear factor
2. Find the matrix representing this transformation.
Solution
a
b
Let the matrix of transformation be .
c
d
A B C A1 B1
C1
a
b
Therefore, 2
2 3 = 4
8 5
c
d
1
3 1
1
3 1
Multiplying these matrices gives,
2a
b 2a
3b
3a
b 4
8 5
=
2c
d 2c
3d
3c
d 1
3 1
Equating the corresponding elements in the two matrices gives:
2a + b = 4 ………..(i)
2c + d = 1 ………….(iv)