Suppose I would like to use Gauss-Jordan method to develop (in matrix form) Cramer's rule for two equations in two unknows:
\begin{pmatrix}a & b & e\cr c & d & f\end{pmatrix}
\begin{pmatrix}1 & \frac{b}{a} & \frac{e}{a}\cr 0 & 1 & \frac{f-\frac{ce}{a}}{d-\frac{bc}{a}}\end{pmatrix}
\begin{pmatrix}1 & 0 & \frac{e}{a}-\frac{b\left( f-\frac{ce}{a}\right) }{a\left( d-\frac{bc}{a}\right) }\cr 0 & 1 & \frac{f-\frac{ce}{a}}{d-\frac{bc}{a}}\end{pmatrix}
\begin{pmatrix}1 & 0 & -\frac{bf-de}{ad-bc}\cr 0 & 1 & \frac{af-ce}{ad-bc}\end{pmatrix}
(In this example) How do I prove that the final matrix represents the correct solution when a is zero regardless of division by a in the first transformation?
I.e. (and this is the more general question):
- how do I keep track of equations ~poisoned by possible zero division (for example, the first row of the second matrix represents an equation which clearly consitst of undefined components when
ais zero)?
(Please excuse my frivolous language :)