I've been working on a problem in which I routinely need to check whether a system of 4 homogeneous linear equations has a unique (zero) solution, or infinite. As of now, I'm doing it by calculating the determinant of the associated matrix.
Out of curiosity I was wondering, not being very proficient in mathematics: are there other ways to determine whether equations are independent or not? Possibly quicker than calulating the determinant, since I don't actually need any more information about the solution(s)?
Usually, the coefficients I'm working with make it impossible to just intuitively check for dependency.
For larger $n$, the Gaussian elimination method requires less operations. (For manual computation, Bareiss might be more attractive).
For $n=4$, that makes little difference. Standard computation involves $24$ terms that are product of $4$ factors.