Compute the RREF of the following matrix :$$\begin{bmatrix}1&-1&2&-3&7\\4&0&3&1&9\\2&-5&1&0&-2\\3&-2&-2&10&-12\end{bmatrix}$$
My friend and I were in pursuit of finding the best way to compute the RREF. We discovered two ways of doing it, and we are trying to find out which one's the most efficient.
Method 1: Bruteforce
This method basically revolves around the very definition of RREF: all the pivots are $1$ and are the only non-zero entry of their column. This would involve annihilating all non-pivot elements of a row by methodically scaling rows and adding/subtracting them(row operations). For example in the above matrix, one would proceed like this: $R_2\to R_2-4R_1,R_3\to R_3-2R_1,R_4\to R_4-3R_1,\cdots$. This results in higher digit numbers sometimes which lead to more complexity in subsequent row operations, since we'd have to work with more digits.
Method 2: Making big numbers small
This method aims to reduce the relatively large elements in a row by performing suitable row operations. For example, instead of performing the above, one would first reduce the $-12$ by performing $R_4\to R_4+R_1$ and then perform $R_2\to R_2-R_4$ which gives $R_2: \{0,3,3,-6,21\}\to\{0,1,1,-2,7\}$. This form and the successive iterations appear much easier to deal with.
Comparision
The first method seems to be very straightforward and algorithmic. The second method seems much more workable and easier to do by hand. However, it appears that the second method takes a larger number of row operations to arrive at the same answer.
There seems to be a tradeoff between arithmetic complexity(crunching ugly, large numbers) and the row operations.
In general, for an arbitrary $m\times n$ matrix, which method is better?
