J20

2a + 3b = 8 ………(ii)
3a + b = 5 …………(iii)
2c + 3d = 3 ……….(v)
3c + d = 1 … ……….(iv)
Solving equations (i) and (ii) simultaneously gives, b = 2 and a = 1.
Solving equations (iv) and (v) simultaneously gives, d = 1 and c = 0.
a
b
1
2
Therefore, =
c
d 0
1
1
2
The matrix represents a shear of factor 2 with the x-axis invariant.
0
1
Example
A line PQ, in which P(10, 4) and Q(2, 8), is mapped onto the line P1Q1, such that P1(5,
2) and Q1(1, 4) after an enlargement of scale factor ½ with centre O. Determine the
matrix representing this transformation.
Solution
a
b
Let the matrix of transformation be
c
d
a
b
10
2
5
1
=
c
d 4 8 2
4
Multiplying these matrices and equating the corresponding elements gives,
10a + 4b = 5 ….(i)
2a + 8b = 1 ……(ii)
10c + 4d = 2 …..(iii)
2c + 8d = 4 …..(iv)
Solving equations (i) and (ii) simultaneously gives, a = ½, b = 0,
Solving equations (iii) and (iv) simultaneously gives c = ½ and d = 0.
a
b
1
0
The matrix = 2
c
d
1
0
2
1
0
Therefore 2 represents an enlargement of scale factor ½ about the origin.
1
0
2
Describing a matrix using the points I(1, 0) and J(0, 1).
It is possible to describe a transformation in matrix form by considering the effect on the
points I(1, 0) and J(0, 1).
1
0
We let I = and J = .
0
1
The columns of a matrix give us the images of I and J after the transformation.
Example
0 1
Describe the transformation with matrix .
1
0