Problem:
Compute the rank of
$\left(
\begin{array}{cc}
 \frac{13}{2} & -2 \\
 10 & -7 \\
 -\frac{19}{2} & 3 \\
\end{array}
\right)$.
Answer:
\begin{array}{l}
 
\begin{array}{l}
 \text{Find the rank of the matrix}: \\
 M=\left(
\begin{array}{cc}
 \frac{13}{2} & -2 \\
 10 & -7 \\
 -\frac{19}{2} & 3 \\
\end{array}
\right) \\
\end{array}
 \\
\hline
 
\begin{array}{l}
 \text{Reduce }\text{matrix }M \text{to }\text{row }\text{echelon }\text{form}: \\
 \left(
\begin{array}{cc}
 \frac{13}{2} & -2 \\
 10 & -7 \\
 -\frac{19}{2} & 3 \\
\end{array}
\right) \\
\end{array}
 \\
 
\begin{array}{l}
 \text{Swap }\text{row }1 \text{with }\text{row }2: \\
 \left(
\begin{array}{cc}
 10 & -7 \\
 \frac{13}{2} & -2 \\
 -\frac{19}{2} & 3 \\
\end{array}
\right) \\
\end{array}
 \\
 
\begin{array}{l}
 \text{Subtract }\frac{13}{20}\, \times \, \text{(row }1) \text{from }\text{row }2: \\
 \left(
\begin{array}{cc}
 10 & -7 \\
 0 & \frac{51}{20} \\
 -\frac{19}{2} & 3 \\
\end{array}
\right) \\
\end{array}
 \\
 
\begin{array}{l}
 \text{Add }\frac{19}{20}\, \times \, \text{(row }1) \text{to }\text{row }3: \\
 \left(
\begin{array}{cc}
 10 & -7 \\
 0 & \frac{51}{20} \\
 0 & -\frac{73}{20} \\
\end{array}
\right) \\
\end{array}
 \\
 
\begin{array}{l}
 \text{Swap }\text{row }2 \text{with }\text{row }3: \\
 \left(
\begin{array}{cc}
 10 & -7 \\
 0 & -\frac{73}{20} \\
 0 & \frac{51}{20} \\
\end{array}
\right) \\
\end{array}
 \\
 
\begin{array}{l}
 \text{Add }\frac{51}{73}\, \times \, \text{(row }2) \text{to }\text{row }3: \\
 \left(
\begin{array}{cc}
 10 & -7 \\
 0 & -\frac{73}{20} \\
 0 & 0 \\
\end{array}
\right) \\
\end{array}
 \\
 
\begin{array}{l}
 \text{There }\text{are }2 \text{nonzero }\text{rows }\text{in }\text{the }\text{row }\text{echelon }\text{form }\text{of }\text{the }\text{matrix, }\text{so }\text{the }\text{rank }\text{is }2: \\
 \fbox{$
\begin{array}{ll}
 \text{Answer:} &  \\
 \text{} & \text{rank}\left(
\begin{array}{cc}
 \frac{13}{2} & -2 \\
 10 & -7 \\
 -\frac{19}{2} & 3 \\
\end{array}
\right) \text{= }\text{rank}\left(
\begin{array}{cc}
 10 & -7 \\
 0 & -\frac{73}{20} \\
 0 & 0 \\
\end{array}
\right)=2 \\
\end{array}
$} \\
\end{array}
 \\
\end{array}

{'source': 'AMPS', 'file': 'amps/mathematica/linear_algebra/rank_and_nullity_w_steps/3114.txt'}