Refresher - Cramer's Rule

Cramer's Rule

Explicit formulas for 2 × 2 and 3 ×3 systems.

Consider the linear system ${\displaystyle \{\begin{array}{l}ax+by=e\\ cx+dy=f\end{array}}$   which in matrix format is ${\displaystyle \left[\begin{array}{cc}a& b\\ c& d\end{array}\right]\left[\begin{array}{c}x\\ y\end{array}\right]=\left[\begin{array}{c}e\\ f\end{array}\right]}$Then, x and y can be found with Cramer's rule as

${\displaystyle x=\left|\begin{array}{cc}e& b\\ f& d\end{array}\right|/\left|\begin{array}{cc}a& b\\ c& d\end{array}\right|=\frac{ed-bf}{ad-bc}}$
and
${\displaystyle y=\left|\begin{array}{cc}a& e\\ c& f\end{array}\right|/\left|\begin{array}{cc}a& b\\ c& d\end{array}\right|=\frac{af-ec}{ad-bc}}$

The rules for 3×3 are similar. Given ${\displaystyle \{\begin{array}{l}ax+by+cz=i\\ dx+ey+fz=k\\ gx+hy+iz=l\end{array}}$   which in matrix format is
${\displaystyle \left[\begin{array}{ccc}a& b& c\\ d& e& f\\ g& h& i\end{array}\right]}$ ${\displaystyle \left[\begin{array}{c}x\\ y\\ z\end{array}\right]=\left[\begin{array}{c}j\\ k\\ l\end{array}\right]}$

the values of x, y and z can be found as follows:

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