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**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...

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