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
a
is zero)?
(Please excuse my frivolous language :)