Reduced row-echelon form of a matrix with variables I have been staring at this for an hour. How would you reduce such a matrix?
\begin{bmatrix}
  p & 0 & a \\
  b & 0 & 0 \\
  q & c & r 
\end{bmatrix}
$abc\neq0$

Because $a,b,c \ne 0$ then
$$
\left( {\begin{array}{*{20}{c}}
   p & 0 & a  \\
   b & 0 & 0  \\
   q & c & r  \\
\end{array}} \right) \to \left( {\begin{array}{*{20}{c}}
   b & 0 & 0  \\
   q & c & r  \\
   p & 0 & a  \\
\end{array}} \right)\mathop  \to \limits_{ - \frac{p}{b}{\rho _1} + {\rho _3}}^{ - \frac{q}{b}{\rho _1} + {\rho _2}} \left( {\begin{array}{*{20}{c}}
   b & 0 & 0  \\
   0 & c & r  \\
   0 & 0 & a  \\
\end{array}} \right)\mathop  \to \limits^{ - \frac{r}{a}{\rho _3} + {\rho _2}} \left( {\begin{array}{*{20}{c}}
   b & 0 & 0  \\
   0 & c & 0  \\
   0 & 0 & a  \\
\end{array}} \right)\mathop  \to \limits^{\frac{1}{b}{\rho _1},\frac{1}{c}{\rho _2},\frac{1}{a}{\rho _3}} \left( {\begin{array}{*{20}{c}}
   1 & 0 & 0  \\
   0 & 1 & 0  \\
   0 & 0 & 1  \\
\end{array}} \right).
$$
