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 $\det A=1$

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.
