J20
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2a + 3b = 8 ………(ii)<br />
3a + b = 5 …………(iii)<br />
2c + 3d = 3 ……….(v)<br />
3c + d = 1 … ……….(iv)<br />
Solving equations (i) and (ii) simultaneously gives, b = 2 and a = 1.<br />
Solving equations (iv) and (v) simultaneously gives, d = 1 and c = 0.<br />
a<br />
b<br />
1<br />
2<br />
Therefore, = <br />
c<br />
d 0<br />
1<br />
1<br />
2<br />
The matrix represents a shear of factor 2 with the x-axis invariant.<br />
0<br />
1<br />
Example<br />
A line PQ, in which P(10, 4) and Q(2, 8), is mapped onto the line P1Q1, such that P1(5,<br />
2) and Q1(1, 4) after an enlargement of scale factor ½ with centre O. Determine the<br />
matrix representing this transformation.<br />
Solution<br />
a<br />
b<br />
Let the matrix of transformation be <br />
c<br />
d <br />
a<br />
b<br />
10<br />
2<br />
5<br />
1 <br />
= <br />
c<br />
d 4 8 2<br />
4<br />
Multiplying these matrices and equating the corresponding elements gives,<br />
10a + 4b = 5 ….(i)<br />
2a + 8b = 1 ……(ii)<br />
10c + 4d = 2 …..(iii)<br />
2c + 8d = 4 …..(iv)<br />
Solving equations (i) and (ii) simultaneously gives, a = ½, b = 0,<br />
Solving equations (iii) and (iv) simultaneously gives c = ½ and d = 0.<br />
a<br />
b<br />
1<br />
0<br />
The matrix = 2 <br />
c<br />
d <br />
1 <br />
<br />
0<br />
2 <br />
1<br />
0<br />
Therefore 2 represents an enlargement of scale factor ½ about the origin.<br />
1 <br />
<br />
0<br />
2 <br />
Describing a matrix using the points I(1, 0) and J(0, 1).<br />
It is possible to describe a transformation in matrix form by considering the effect on the<br />
points I(1, 0) and J(0, 1).<br />
1<br />
0 <br />
We let I = and J = .<br />
0<br />
1<br />
<br />
The columns of a matrix give us the images of I and J after the transformation.<br />
Example<br />
0 1 <br />
Describe the transformation with matrix .<br />
1<br />
0