Cramer's rule

When the determinant of the coefficients is different from zero, the system has one and only one solution, obtained as a ratio of determinants.

Theorem — Cramer's

If $\Delta \neq 0$, the system has a unique solution: $x = \frac{\Delta_x}{\Delta} = \frac{\begin{vmatrix} c_1 & b_1 \\ c_2 & b_2 \end{vmatrix}}{\Delta}, \qquad y = \frac{\Delta_y}{\Delta} = \frac{\begin{vmatrix} a_1 & c_1 \\ a_2 & c_2 \end{vmatrix}}{\Delta}.$

To obtain $\Delta_x$ we replace, in the coefficient matrix, the column of $x$ with the column of the constant terms; likewise for $\Delta_y$ we replace the column of $y$.

Connections

Topics: Linear systems
Concepts: Determinant · Cramer’s rule
Methods: Systems by Cramer
Skills: Solving systems · Using formulae
People: Gabriel Cramer