Cramer's rule

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Consider a set of n linear equations in n unknowns x1, x2,…, xn, written in matrix form as Ax=b. When A is invertible, the set of equations has a unique solution x=A−1b. Since A−1=(1/det A) adj A, this gives the solutionwhich may be writtenusing the entries of b and the cofactors of A. This is Cramer's rule. Note that here the numerator is equal to the determinant of the matrix obtained by replacing the j-th column of A by the column b. For example, this gives the solution of

ax + by = h,cx + dy = k,

when adbc ≠ 0, as

Subjects: Mathematics.

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