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
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det(A) where Aj is the matrix A whose j th column has been replaced by the constants in B .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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