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**Unformatted text preview: **lutions as follows.
x1 = 3
4
3
4 = −14
4 − 18
=
= 2,
8 − 15
−7 x2 = (6) 1
6
2
5
2
5
2
5 1
6
3
4 = 12 − 5
7
=
= −1.
8 − 15
−7 and
(7) Do you see the pattern? The determinant on the bottom has the coeﬃcients on the left side of the equations,
and the determinants on the top replace the ﬁrst or second column with the numbers on the right side of
the equations. Cramer’s Rule works in all dimensions, but it isn’t any more eﬃcient than any of our other
techniques. It is evidence, however, that the determinant captures something important.
3 × 3 determinants. The determinant of a 3 × 3 matrix can be deﬁned as follows.
(8) a11
a21
a31 a12
a22
a32 a13
a23
a33 = a11 a22 a33 + a12 a23 a31 + a13 a21 a32 − a13 a22 a31 − a12 a21 a33 − a11 a23 a32 . There are patterns that are followed in deﬁning determinants of all orders. One of the relevant patterns here
is that each...

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