516 Chapter 18Matrices and linear transformations
The determinant of this matrix is shown in Example 17.4 to have valueA 1 = 16 , and the
cofactors of several of its elements are computed in Example 17.5. The matrix of the
cofactors is
and the inverse is the transpose of this divided by the value of the determinant,
Check:
0 Exercises 48–52
We note that the above method of finding the inverse of a matrix involves the
evaluation of the determinant and ofn
2cofactors. As discussed in the note to Section
17.3, this is not a practical method except for very small matrices (see Chapter 20).
The inverse of a matrix product
The inverse of the product of two, or more, matrices is equal to the product of the
inverse matrices taken in reverse order,
(AB)
− 11 = 1 B
− 1A
− 1(18.46)
0 Exercise 53
18.5 Linear transformations
The coordinate transformation discussed in Section 18.1, equations (18.3) to (18.7),
is a type of linear transformation. More generally, a linear transformation is a set of
linear equations*
AA
−=
−−
−
−
11
6
036
0312
238
213
420
110
=
=
1
6
600
060
006
100
010
0001
A
−=
−−
11
6
036
0312
238
002
333
612 8
−
−
*The mostgeneral form is a transformation of a vector with ncomponents into a vector with mcomponents.
The transformation matrix is then rectangular with dimensionsm 1 × 1 n.