18.6 Orthogonal matrices and orthogonal transformations 523
and therefore,A
T1 = 1 A
− 1. It also follows that the determinant of an orthogonal matrix
has value±1. Thus, becausedet 1 A
T1 = 1 det 1 Aanddet 1 I 1 = 11 , we have
det 1 (A
TA) 1 = 1 det 1 A
T1 × 1 det 1 A 1 = 1 (det 1 A)
21 = 11 (18.64)
EXAMPLE 18.22 The matrix
is orthogonal with properties
(i)
andAA
T1 = 1 (A
TA)
T1 = 1 I.
(ii) The columns of Aform the vectors
with properties
Similarly for the rows of A(the columns ofA
T).
ab 11 bc 11 ca 11
·· ·=−+=, =−−+=, =−
2
9
4
9
2
9
0
2
9
2
9
4
9
0
4
99
2
9
2
9
++= 0
aa bb cc 11 11 11
···
===
2221
3
2
3
2
3
= 1
ab c=,,
,=,−,
,=−,
2
3
2
3
1
3
1
3
2
3
2
3
2
3
1
3
,,
2
3
AA
T=−
−
2
3
2
3
1
3
1
3
2
3
2
3
2
3
1
3
2
3
−
−
=
2
3
1
3
2
3
2
3
2
3
1
3
1
3
2
3
2
3
1100
010
001
A=
−
−
2
3
1
3
2
3
2
3
2
3
1
3
1
3
2
3
2
3