504 Chapter 18Matrices and linear transformations
The transpose matrix
The transposeA
T(or¡) of anm 1 × 1 nmatrix Ais then 1 × 1 mmatrix obtained from A
by the interchange of rows and columns; the first column ofA
Tis the first row of A,
the second column ofA
Tis the second row of A, and so on. For example,
(18.17)
EXAMPLES 18.3Transpose matrices
(i) if then
(ii) if then
(iii) if then a
T1 = 1 (3 2 −1)
(iv) if then A
T1 = 1 A
0 Exercises 7–12
Case (iv) above is an example of a symmetric matrix, whose transpose is equal to the
matrix:
A
T1 = 1 A (18.18)
The determinant
The value of the determinant of a square matrix is unchanged when the matrix is
transposed,
det 1 A
T1 = 1 det 1 A (18.19)
A=−
−
130
321
010
a=
−
3
2
1
A
T=
−
13
24
A=
−
12
34
A
T=
−
11
24
02
A=
−
120
142
if AA==then
ab
ab
ab
a
1
122331
Taaa
bbb
23123