Cramer's Rule

Cramer's Rule is a method for solving multivariate simultaneous linear equations.

a₁₁ x₁ + a₁₂x₂ + ... a_1n x_n = b₁
a_n1 x₁ + a_n2 x₂ + ... a_nnx_n = b_n

where x_i is the ith variable, a_ij is the constant coefficient on *y in the rth equation, and b_i is the constant on the right-hand side of the rth equation.

This can be written in matrix notation as:

AX = b,

where A is the matrix containing the elements a_ij b is the vector containing the elements b_i ; X is the vector of values of the variables x_i The value of x_k which satisfies. the set of simultaneous equations is found by Cramer's rule by replacing the kth column of the matrix A by the vector b, forming a new matrix A_k.

The value of X_k is then the determinant of Ak divided by the determinant of A, that is

X_k = |A_k| / |A|; k=1,...,n

Source:
K A Fox and T K Kaul, Intermediate Economic Statistics (Melbourne, Fla, 1980)

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