530 Chapter 18Matrices and linear transformations
Find the commutator of the following pairs of matrices:
42.The spin matrices for a nucleus with spin quantum number 1 are
(i)Find the commutators , ,. (ii)Find
43.Given the matrix , find matrices Band Cfor whichBA 1 = 1 AC 1 = 1 A.
Find the general matrix for which:
46.Given and , show that(AB)
T
= 1 B
T
A
T
Section 18.4
Find the inverse matrix, if possible:
52.For each matrix (A) of Exercises 47–51, verify (ifA
− 1
exists) thatAA
− 1
1 = 1 A
− 1
A 1 = 1 I.
53.If and , show that (AB)
− 1
1 = 1 B
− 1
A
− 1
Section 18.5
54.The linear transformationr′ 1 = 1 Ar, where
rrA′=
′
′
′
,=
,=
x
y
z
x
y
z
coos sin
sin cos
θθ
θθ
−
0
0
001
B=
56
78
A=
12
34
3400
1200
0031
0042
1
2
1
2
1
2
1
2
0
01 0
0
−
211
112
310
−
−−
−
123
212
311
−
−−
23
41
−
B=
−
12
04
A=
−
−
−
53
41
21
12
24
12
24
=
ab
cd
ab
cd
=
00
00
12
24
00
00
=
ab
cd
ab
cd
A=
1234
5678
III
xyz222
II ++.
zx,
II
yz,
II
xy,
II
xyi
ii
i
=
,=
−
−
2
010
101
010
2
00
0
00
,=
−
I
z10 0
00 0
00 1
10
10
01
10
,
−
21
11
13
32
−
−
,
−