Solving a system of linear equations using Cramer's rule

Cramer's rule may only be applied for a system of linear equations with as many equations as unknowns (the coefficient matrix of the system must be square) and with non-zero determinant of the coefficient matrix.

Consider a system of n linear equations for n unknowns x₁, x₂, ..., x_n:
a₁₁x₁ + a₁₂x₂ + ...+ a_1nx_n = b₁
a₂₁x₁ + a₂₂x₂ + ...+ a_2nx_n = b₂
   ...   ...   ...    ...    ...
a_n1x₁ + a_n2x₂ + ...+ a_nnx_n = b_n

The determinant of the coefficient matrix



Let be the determinant of the matrix formed by replacing the j column with the column of the constant terms

If , the system has a unique solution

Since the computation of large determinants is cumbersome, Cramer's rule is generally used for systems of two and three equations.