Cramer's Rule

Discipline: Economics

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

a11 x1 + a12x2 + ... a1n xn = b1
an1 x1 + an2 x2 + ... annxn = bn

where xi is the ith variable, aij is the constant coefficient on *y in the rth equation, and bi 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 aij b is the vector containing the elements bi ; X is the vector of values of the variables xi The value of xk 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 Ak.

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

Xk = |Ak| / |A|; k=1,...,n

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


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