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Cramer’s Rule
To solve system (3), we multiply the first equation by À11, the second by À21, etc., the last equation is multiplied by Àn1. Then, we sum the equations and collect similar terms:
Consider the nth-order determinant composed of the coefficients of system (3):
The coefficient of õ1 is the sum of the products of the element of the first column and their algebraic complements. According to property 9, it equals determinant (4). The coefficients of the unknowns The right-hand side is the product of the free terms and the algebraic complements of the elements of the first column; consequently, it equals the determinant (4) in which the first column is replaced by the column of free terms:
Expressions for the other unknowns are obtained in a similar way: we multiply system (3) by the algebraic complements of the n corresponding columns
where Example. Solve the system of equations Let us evaluate the principal determinant of the system:
To obtain zeros in the first row, we leave the third column unchanged; multiply it by –2 and add to the first column; then multiply it by 3 and add to the second column. Let us calculate the auxiliary determinants. õ1 is derived from by replacing the first column by the free terms:
By Cramer's rule (5) we obtain
1. In (5), the principal determinant must be different from zero. In this case, system (3) has a unique solution. 2. If 3. If
Date: 2015-01-02; view: 1360
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