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5 even though a system of equations wasnt formally

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Unformatted text preview: s x, y , z , etc. The proof of the general case is best left to a course in Linear Algebra. 3 Essentially, we follow the Gauss Jordan algorithm but we don’t care about getting leading 1’s. As we will see in Section 8.5.2, determinants (specifically cofactors) are deeply connected with the inverse of a matrix. 4 8.5 Determinants and Cramer’s Rule 513 Theorem 8.8. Cramer’s Rule: Suppose AX = B is the matrix form of a system of n linear equations in n unknowns where A is the coefficient matrix, X is the unknowns matrix, and B is the constant matrix. If det(A) = 0, then the corresponding system is consistent and independent and the solution for unknowns x1 , x2 , . . . xn is given by: xj = det (Aj ) , det(A) where Aj is the matrix A whose j th column has been replaced by the constants in B . In words, Cramer’s Rule tells us we can solve for each unknown, one at a time, by finding the ratio of the determinant of Aj to that of the determinant of the coefficient matrix. The matrix Aj is found by replacing the column in the coefficient matrix which holds the coefficients of xj with the constants of the system. The...
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