Cramer's rule

Cramer's Rule is a mathematical theorem used to solve systems of linear equations with as many equations as unknowns, provided that the system has a unique solution. It is applicable when the coefficient matrix is non-singular (i.e., its determinant is non-zero).

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A

and

A x = b

, replace column

i

b

to form

A_{i}

. Multilinearity of the determinant and

b = \sum_{j} x_{j} A_{\cdot j}

give

det A_{i} = x_{i} det A

: all other summands have repeated columns and vanish. Division gives the formula. With integer coefficients this supplies explicit rational solutions and, together with the Hadamard determinant inequality, bounds their numerators and denominators.