Unformatted text preview: is the product of A with the
second column of X and the third column of I3 is the product of A with the third column of X .6
In other words, we are solving three equations x11
1
x12
0
x13
0
A x21 = 0 A x22 = 1 A x23 = 0 x31
0
x32
0
x33
1
We can solve each of these systems using Cramer’s Rule. Focusing on the ﬁrst system, we have 1
1
2
312
3
1
1
5 A2 = 0 0 5 A3 = 0 −1
0
A1 = 0 −1
0
1
4
204
2
1
0
5
We are developing a method in the forthcoming discussion. As with the discussion in Section 8.4 when we
developed the ﬁrst algorithm to ﬁnd matrix inverses, we ask that you indulge us.
6
The reader is encouraged to stop and think this through. 8.5 Determinants and Cramer’s Rule 515 If we expand det (A1 ) along the ﬁrst row, we get
det (A1 ) = det −1 5
14 = det −1 5
14 − det 05
04 + 2 det 0 −1
0
1 Amazingly, this is none other than the C11 cofactor of A. The reader is invited to check this, as
well as the claims that det (A2 ) = C12 and det (A3 ) = C13 .7 (To see this, though it seems unnatural
to do so, expand along the ﬁr...
View
Full Document
 Fall '13
 Wong
 Algebra, Trigonometry, Cartesian Coordinate System, The Land, The Waves, René Descartes, Euclidean geometry

Click to edit the document details