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M1410805

# M1410805 - (Section 8.5 Applications of Determinants 8.67...

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(Section 8.5: Applications of Determinants) 8.67 SECTION 8.5: APPLICATIONS OF DETERMINANTS PART A: CRAMER’S RULE FOR SOLVING SYSTEMS A square system of linear equations is a system of n linear equations in n unknowns, where n Z + . Cramer’s Rule uses determinants to solve such a system. For now, we assume that the unknowns are x , y , etc. and that they make up X , the vector of unknowns. Cramer’s Rule Write the augmented matrix for the system AX = B : A B A is the coefficient matrix. If the system is square, A will be a square matrix. B is the right-hand side ( RHS ); you could use RHS , instead. Compute the following determinants: • Let D = A , or det A ( ) . • Let D x = A x , or det A x ( ) . where A x is identical to A , except that the x -column of A is replaced by B , the RHS . (continued on next page)

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(Section 8.5: Applications of Determinants) 8.68 Cramer’s Rule (cont.) • Let D y = A y , or det A y ( ) , where A y is identical to A , except that the y -column of A is replaced by B , the RHS . D z , A z , etc. are defined analogously as necessary. If D 0 , there is a unique solution given by: x = D x D , y = D y D , z = D z D (if applicable), etc. If D = 0 , there is not a unique solution. Then: • If all of the other determinants, D x , D y , etc. are also 0, then the system has infinitely many solutions. • Otherwise, the system has no solution. The solution set is , the empty set. Note: Observe that the formulas for x , y , etc. fall apart if D = 0 . Note: In fact, if A is square, then its determinant D 0 if and only if A is invertible, which is true if and only if AX = B has a unique solution (given by X = A 1 B ). See the Inverse Matrix Method for solving systems in Section 8.3, Part D . Note: One advantage that this method has over Gaussian Elimination with Back-
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M1410805 - (Section 8.5 Applications of Determinants 8.67...

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