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Unformatted text preview: Conversely, suppose that det A = 0. Then the homogeneous system AX = 0
has a non–trivial solution X = [x1 , . . . , xn ]t . So
x1 A∗1 + · · · + xn A∗n = 0.
Suppose for example that x1 = 0. Then
− A ∗1 = x2
x1 + ··· + − xn
x1 A ∗n and the ﬁrst column of A is a linear combination of the remaining columns.
14. Consider the system
−2x + 3y − z =
1
x + 2y − z =
4
−2x − y + z = −3
0 7 −3
−2
3 −1
7 −3
1
2 −1 = 1 2 −1 = −
= −2 = 0.
Let ∆ =
3 −1
0 3 −1
−2 −1
1
Hence the system has a unique solution which can be calculated using
Cramer’s rule:
∆2
∆3
∆1
, y=
, z=
,
x=
∆
∆
∆
where
∆1 = ∆2 = ∆3 =
Hence x = −4
−2 = 2, y = −6
−2 1
3 −1
4
2 −1
−3 −1
1 −2
1 −1
1
4 −1
−2 −3
1 −2
3
1
1
2
4
−2 −1 −3 = 3, z = −8
−2 = −4,
= −6,
= −8. = 4. 15. In Remark 4.0.4, take A = In . Then we deduce
(a) det Eij = −1;
(b) det Ei (t) = t;
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This note was uploaded on 12/19/2011 for the course MAS 3105 taught by Professor Dreibelbis during the Fall '10 term at UNF.
 Fall '10
 Dreibelbis

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