18.8 Exercises 529
representations’) are distinct and independent. In the present case, all the possible
representations of the group Gare either equivalent to or may be reduced to the three
representations in Table 18.2.
0 Exercise 63
18.8 Exercises
Section 18.1
1.Construct transformation matrices that represent the following rotations about the z-axis:
(i)anticlockwise through 45°, (ii)anticlockwise through 90°,
(iii)clockwise through 90°.
2.Construct a transformation matrix that represents the interchange ofxand ycoordinates
of a point.
Section 18.2
For the following matrices,
find, if possible:
3.det 1 A, tr 1 A4.det 1 D, tr 1 D5.det 1 P, tr 1 P6.det 1 a, tr 1 a7.A
T
- C
T
- D
T
- P
T
- a
T
- b
T
Section 18.3
For the above matrices find, if possible:
- A 1 + 1 B 14. A 1 − 1 B 15. B 1 − 1 A 16. C 1 + 1 D 17. a 1 + 1 b
T
- a
T
1 + 1 b
- 3 P 20. 2 A 1 + 13 B 21. AB 22. BC 23. CB 24. CP
- PC 26. D
2
- PQ 28. QP 29. Ba 30. ab
- ba 32. a
T
b
T
- b
T
a
T
- Ca 35. a
T
C
36.If , show thatA
3
1 − 1 A
2
1 − 13 A 1 + 1 I 1 = 10.
37.If and , find Asuch that
Given the matrices Aand B, findABandBA:
- AB=
,=
−
10
10
01
10
AB=
−
−
,=
−
21
11
13
32
AB=
32
14
B=
21
32
A=
ab
cd
A=
−−
11 1
21 2
21 1
PQa=
−
=
=−
12
04
30
01
0
3
1
b=−()25 2
AB C=
−
=
−
−
=
−
−
123
034
014
230
53
41
221
30 0
02 0
− 00 1
=
−
D