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:
$\left[\begin{array}{ccc}a& b& c\\ d& e& f\\ g& h& i\end{array}\right]\left[\begin{array}{c}x\\ y\\ z\end{array}\right]=\left[\begin{array}{c}j\\ k\\ l\end{array}\right]$

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

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