“Cramer's Rule”的概念、定义、习题解题方法及翻译-数学辞典

单词

Cramer's Rule

释义

Cramer's Rule

Given a set of linear equations

(1)

consider the Determinant

(2)

Now multiply

, and use the property of Determinants that Multiplication by aconstant is equivalent to Multiplication of each entry in a given row by that constant

(3)

Another property of Determinants enables us to add a constant times any column to any column andobtain the same Determinant, so add

times column 2 and

times column 3 to column 1,

(4)

, then (4) reduces to

, so the system has nondegenerate solutions (i.e., solutions other than(0, 0, 0)) only if

(in which case there is a family of solutions). If

and

, thesystem has no unique solution. If instead

and

, then solutions are given by

(5)

and similarly for

			(6)
			(7)

This procedure can be generalized to a set of equations so, given a system of linear equations

(8)

let

(9)

, then nondegenerate solutions exist only if

. If

and

, the system has no unique solution. Otherwise, compute

(10)

Then

for

. In the 3-D case, the Vector analog of Cramer's rule is

(11)

See also Determinant, Linear Algebra, Matrix, System of Equations, Vector

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