Question

Construct the 2 × 2 matrix for the linear transformations R 2 → R 2 defined by the following compositions. In each case, write down the matrix of each transformation, then multiply the matrices in the correct order.

(a) A dilation by a factor of 4, then a reflection across the x-axis.

(b) A counterclockwise rotation through π/2 , then a dilation by a factor of 1/2 .

(c) A reflection about the line x = y, then a rotation though an angle of π.

Answer 1

(a). The matrix representing dilation by a factor of 4 is M =

4	0
0	4

The matrix representing reflection across the X-Axis is N =

1	0
0	-1

Hence, the matrix representing dilation by a factor of 4 and then reflection across the X-Axis is NM =

4	0
0	-4

b). The matrix representing counterclockwise rotation through π/4 is M =

cos π/4	-sin π/4
sin π/4	cos π/4

=

1/√2	-1/√2
1/√2	1/√2

   The matrix representing dilation by a factor of 1/2 is N =

½	0
0	½

Hence, the matrix representing counterclockwise rotation through π/4 and then dilation by a factor of 1/2 is NM =

1/(2√2)	-1/(2√2)
3/(2√2)	-1/(2√2)

(c ). The matrix representing reflection across the line x = y is M =

0	1
1	0

The matrix representing counterclockwise rotation through π is N =

cos π	-sin π
sin π	cos π

=

-1	0
0	-1

Hence, the matrix representing reflection across the line x = y and then counterclockwise rotation through π is NM =

0	-1
-1	0