# Check if a homogeneous system of linear equations has infinite solutions (not solving it) without calculating a determinant.

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.

• If your equations are over a finite field (for coding, say), there are only finitely many solutions anyway without calculating anything. Commented May 9, 2019 at 13:54
• Determinants are a poor way to test for dependency - the calculation can be numerically unstable. A better strategy is to calculate a row reduced form for the coefficient matrix. That's standard linear algebra. en.wikipedia.org/wiki/Gaussian_elimination Commented May 9, 2019 at 13:55

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.