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 $\det A=1$
 A: I suggest Gaussian Elimination till upper triangle form or further but keep track of the effect of each elementary.
see here for elementary's effect on det
A: 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...
A: For a 4x4 determinant I would probably use the method of minors: the 3x3 subdeterminants have a convenient(ish) mnemonic as a sum of products of diagonals and broken diagonals, with all the diagonals in one direction positive and all the diagonals in the other direction negative; this lets you compute the determinant of e.g. the bottom-right 3x3 as 71*73*38 + 78*32*50 + 34*69*65 - 34*73*50 - 71*32*65 - 78*69*38.  That's probably slightly less than a 5-minute calculation with pencil and paper and a 1-minute calculation with a calculator, which means you could find the overall determinant in maybe 5 minutes with calculator, 15-20 with pencil and paper.  Not blazingly fast, of course, but for me I suspect it'd be marginally faster than Gaussian Elimination, and the all-integer nature of it is (for me, at least) a minor plus.  Alternately, the subdeterminants can be computed by taking minors again; this cuts down slightly on the number of multiplications per subdeterminant(from 12 to 9) and gives a total of 40 multiplications to compute the 4x4 determinant.
