Cramer's Rule

Cramer’s Rule is a way to solve a system of equations using determinants of the function’s coefficients. It is useful for solving differential equations, particularly of nonlinear order using the method of variation of parameters

Workflow

Solving a system of equations with coefficients $a,b,c,d,...$

$$\{\begin{array}{l}ax\_1+bx\_2+...=z\_1\\ cx\_1+dx\_2+...=z\_2\end{array}$$

Cramer’s rule said that the solution to the unknowns (here $x\_1\text{and}x\_2$) is given by the formula

$$x\_n=\frac{|A\_n|}{|A|}$$

where $|A|$ is the determinant of the matrix formed by plotting the unknown’s coefficients into a matrix

$$A=\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]$$

here, the first column represents all coefficients of $x\_1$ and the second of $x\_2$.

The determinant matrix $A\_n$ is a modified matrix $A$, in which the column that contains the coefficients to the unknown we want to find has been switched for the column of right-hand values (here $z\_1,z\_2$) of the system. Solving for $x\_1$ one would replace the first column of $A$ to retrieve

$$A\_n=\left[\begin{array}{cc}z\_1& b\\ z\_2& d\end{array}\right]$$

such that $x\_1$ is found with

$$x\_1=\frac{|A\_n|}{|A|}=\frac{det(\left[\begin{array}{cc}z\_1& b\\ z\_2& d\end{array}\right])}{det(\left[\begin{array}{cc}a& b\\ c& d\end{array}\right])}$$