Problem:
Find the determinant of the matrix
$\left(
\begin{array}{cc}
 \frac{9}{2} & -\frac{7}{2} \\
 -\frac{25}{6} & -\frac{1}{6} \\
\end{array}
\right)$.
Answer:
\begin{array}{l}
 
\begin{array}{l}
 \text{Find the determinant}: \\
 \left| 
\begin{array}{cc}
 \frac{9}{2} & -\frac{7}{2} \\
 -\frac{25}{6} & -\frac{1}{6} \\
\end{array}
\right|  \\
\end{array}
 \\
\hline
 
\begin{array}{l}
 \text{Multiply }\text{along }\text{the }\text{diagonals }\text{and }\text{subtract}: \\
 \frac{1}{6} \left(-\frac{9}{2}\right)-\left(-\frac{7}{2}\right)\, \left(-\frac{25}{6}\right) \\
\end{array}
 \\
 
\begin{array}{l}
 \frac{9}{2}\times \frac{-1}{6}=\frac{9 (-1)}{2\times 6}: \\
 \frac{-9}{2\times 6}-\frac{-7}{2}\times \frac{-25}{6} \\
\end{array}
 \\
 
\begin{array}{l}
 \text{The }\text{gcd }\text{of }9 \text{and }6 \text{is }3, \text{so }\frac{9 (-1)}{2\times 6}=\frac{(3\times 3) (-1)}{2 (3\times 2)}=\frac{3}{3}\times \frac{3 (-1)}{2\times 2}=\frac{3 (-1)}{2\times 2}: \\
 \frac{-\fbox{$3$}}{2\times \fbox{$2$}}-\frac{-7}{2}\times \frac{-25}{6} \\
\end{array}
 \\
 
\begin{array}{l}
 2\times 2 \text{= }4: \\
 \frac{-3}{4}-\frac{-7}{2}\times \frac{-25}{6} \\
\end{array}
 \\
 
\begin{array}{l}
 -(-7) \text{= }7: \\
 \frac{-3}{4}+\frac{7 (-25)}{2\times 6} \\
\end{array}
 \\
 
\begin{array}{l}
 7 (-25) \text{= }-175: \\
 \frac{-3}{4}-\frac{175}{2\times 6} \\
\end{array}
 \\
 
\begin{array}{l}
 2\times 6 \text{= }12: \\
 \frac{-3}{4}-\frac{175}{12} \\
\end{array}
 \\
 
\begin{array}{l}
 -\frac{3}{4}-\frac{175}{12}=-\frac{46}{3}: \\
 \fbox{$
\begin{array}{ll}
 \text{Answer:} &  \\
 \text{} & \frac{-46}{3} \\
\end{array}
$} \\
\end{array}
 \\
\end{array}
{'source': 'AMPS', 'file': 'amps/mathematica/linear_algebra/determinant_w_steps/4313.txt'}