ECON 211 Chapter 5: Cramer's Rule

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27 Jul 2016
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Economics 211
Cramer’s Rule
.
Consider the system of linear equations Ax =bwhere Ais nby n, and xand bare nby
1 column vectors. If the determinant of Ais not zero we have seen that the solution for xis
A1bwhere
A1=1
det(A)
c11 c21 . . . cn1
c12 c22 . . . cn2
.
.
..
.
..
.
.
c1nc2n. . . cnn
,
A= [aij ] and cij is the cofactor of aij . Thus the ith row of the solution vector xis
xi=1
det(A)(c1ib1+c2ib2+· · · +cnibn).(1)
Cramer’s Rule says that xi= det(A0)/det(A) where A0is Awith the ith column replaced
by b, that is,
xi=1
det(A)det
a11 . . . b1. . . a1n
a21 . . . b2. . . a2n
.
.
..
.
..
.
.
an1. . . bn. . . ann
.
Using the ith column to calculate det(A0) we obtain
xi=1
det(A)(b1c1i+b2c2i+· · · +bncni),
which is the same as (1).
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