Advanced Microeconomics/Linear Algebra

Cramer's Rule
For a three by three system:

$$\begin{array}{c} ax+by+cz=j\\ dx+ey+fz=k\\ gx+hy+iz=l\\ \end{array} $$ In matrix form: $$ \begin{bmatrix} a&b&c\\d&e&f\\g&h&i \end{bmatrix} \begin{bmatrix} x\\y\\z \end{bmatrix} = \begin{bmatrix} j\\k\\l \end{bmatrix} $$

$$x = \frac{\begin{vmatrix}j&b&c\\k&e&f\\l&h&i\end{vmatrix}}{\begin{vmatrix}a&b&c\\d&e&f\\g&h&i\end{vmatrix}} $$ $$y = \frac{\begin{vmatrix}a&j&c\\d&k&f\\g&l&i\end{vmatrix}}{\begin{vmatrix}a&b&c\\d&e&f\\g&h&i\end{vmatrix}} $$ $$z = \frac{\begin{vmatrix}a&b&j\\d&e&k\\g&h&l\end{vmatrix}}{\begin{vmatrix}a&b&c\\d&e&f\\g&h&i\end{vmatrix}} $$

In general: $$Ax = b$$ $$ x_i = \frac{|A_i|}{|A|} \; \forall i= \{1,\dots,n\}$$ Where $$A_i$$ is formed by replacing the vector associated with $$x_i$$ by the column vector b.