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.