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, ..., xn:
a11x1 + a12x2 + ...+ a1nxn = b1
a21x1 + a22x2 + ...+ a2nxn = b2
   ...   ...   ...    ...    ...
an1x1 + an2x2 + ...+ annxn = bn

The determinant of the coefficient matrix

determinant of the coefficient matrix

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

If determinant nonzero, the system has a unique solution Cramer's rule

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

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