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**Unformatted text preview: **ltimate sign of the term in the expansion of the determinant.
To illustrate some of the other properties in Theorem 8.7, we use row operations to transform our
3 × 3 matrix A into an upper triangular matrix, keeping track of the row operations, and labeling
2
For a very elegant treatment, take a course in Linear Algebra. There, you will most likely see the treatment of
determinants logically reversed than what is presented here. Speciﬁcally, the determinant is deﬁned as a function
which takes a square matrix to a real number and satisﬁes some of the properties in Theorem 8.7. From that function,
a formula for the determinant is developed. 512 Systems of Equations and Matrices each successive matrix.3 3
1 0 −1
2
1
A 2
3
1
Replace R3
5 − − − − − → 0 −1
−−−−−
with − 2 R1 + R3
1
3
4
0
3
B 2
3
1
Replace R3 with
5 − − − − − → 0 −1
−−−−−
1
R2 + R3
8
3
0
0
3
C 2
5 13
3 Theorem 8.7 guarantees us that det(A) = det(B ) = det(C ) since we are replacing a row with
itself plus a mu...

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