Cramer's rule

Cramer’s rule is a formula for the solution of a system of linear equations with $n$ equations and $n$ unknowns, whenever the system has a unique solution.

For $2\times 2$ matrices, Cramer’s rule forms a competent alternative to Gaussian elimination (assuming there’s a unique solution). For matrices with a larger size, Gaussian elimination is more computationally efficient, due to Cramer’s rule’s reliance on determinants.

The $2\times 2$ case is as follows. For a system of 2 equations with 2 unknowns:

$$a{x}_1+b{x}_2=e$$ $$c{x}_1+d{x}_2=f$$

Then Cramer’s rule states that:

$$x_1=\frac{\left|\begin{array}{cc}e& b\\ f& d\end{array}\right|}{\left|\begin{array}{cc}a& b\\ c& d\end{array}\right|}$$ $$x_2=\frac{\left|\begin{array}{cc}a& e\\ c& f\end{array}\right|}{\left|\begin{array}{cc}a& b\\ c& d\end{array}\right|}$$

Note our observation from above, where the system must have a unique solution. Because the determinant of the denominators cannot be 0, the system must be linearly independent.