How to efficiently compute the determinant of a matrix using elementary operations? Need help to compute $\det A$ where
$$A=\left(\begin{matrix}36&60&72&37\\43&71&78&34\\44&69&73&32\\30&50&65&38\end{matrix} \right)$$
How would one use elementary operations to calculate the determinant easily?
I know that $\det A=1$

Here's one way to do it without fractions.
You could start by subtracting row $2$ from row $3$ to get
$$ \left[ \begin {array}{cccc} 36&60&72&37\\ 43&71&78&34\\ 1&-2&-5&-2\\ 30&50&65&38
\end {array} \right]$$
Then subtract $36$, $43$, and $30$ times row $3$ from rows $1$, $2$ and $4$ respectively to get 
$$ \left[ \begin {array}{cccc} 0&132&252&109\\ 0&157&293&120\\ 1&-2&-5&-2\\ 0&110&215&
98\end {array} \right]$$
Expanding by minors in the first column, we just need one $3 \times 3$ determinant, which is $$132 \times 293 \times 98 + 252 \times 120 \times 110 + 109 \times 157 \times 215 - 132 \times 120 \times 215 - 252 \times 157 \times 98 - 109 \times 293 \times 110 = 1$$
I hope you're allowed to use a calculator for that...
