Cramer's Rule

Cramer’s rule is a formula for the solution of a set of linear equations. It’s mostly used for a small systems ($n=2,n=3$), as higher numbers becomes computationally expensive/complex to implement.

The system is formulated in matrix form:

$$\begin{array}{rl}[A]\{x\}& =\{b\}\\ \left[\begin{array}{ccc}{a}_{11}& a_{12}& a_{13}\\ a_{21}& a_{22}& a_{23}\\ a_{31}& a_{32}& a_{33}\end{array}\right]\left\{\begin{array}{c}{x}_1\\ x_2\\ x_3\end{array}\right\}& =\left\{\begin{array}{c}{b}_1\\ b_2\\ b_3\end{array}\right\}\end{array}$$

Cramer’s rule then states:

$$x_1=\frac{\left|\begin{array}{ccc}{b}_1& a_{12}& a_{13}\\ b_2& a_{22}& a_{23}\\ b_3& a_{32}& a_{33}\end{array}\right|}{D}$$

where $D$ is the Determinant of $[A]$.

Basically, you replace column 1 values of $A$ with $\left\{\begin{array}{c}B\end{array}\right\}$, calculate the determinant of the top and use the determinant of $A$ on the bottom.

• For $x_2$, you would replace column 2 of $A$ with $\{B\}$
• For $x_3$, you would replace column 3 of $A$ with $\{B\}$