Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

If I have a system of N linear equations with N unknowns, how do I know if each unknown has atleast one solution. I havn't taken any formal classes in linear algebra, so please if you use any abstract terminology associated with linear algebra, explain it please. I would appreciate any help.

share|cite|improve this question
This is a useful method: I don't think there is any simple way of going around linear algebra on a question like this (assuming that your N equations are of linear form). – T. Eskin Nov 27 '12 at 9:23
I apologise I should have reworded my question better. I don't need help solving a system of linear equations, I want to know under what condinitions a system with n unknowns and n equations IS solveable. I am also not asking you to not use linear algebra, I am just asking you to explain any abstract notation, if you chose to use it. – Ethan Nov 27 '12 at 9:26
Are we to assume these are linear equations? I think in general knowing in advance whether we can solve any set of $N$ equations in $N$ unknowns is very difficult, if not impossible. However if they are linear equations then there is a very simple test, calculate the determinant ( If it is non-zero (which is extremely likely) then you have a unique solution. If it is $0$ then you either have no solutions or infinitely many, and it is easy to check which is the case. – fretty Nov 27 '12 at 9:32
Yes all the equations are linear, and I don't even need unique solutions, I just need to know that all N unknowns have atleast one solution. Also I don't know what a "determinant" is, if you could help me out, id appreciate it. – Ethan Nov 27 '12 at 9:38
no you can't always find solution, the coefficients must satisfy certain conditions – user31280 Nov 27 '12 at 10:09
up vote 0 down vote accepted
  • Take a look at this link explaining where a certain matrix is proven to be non invertivible using a system of linear equations.
  • Also read this Wikipedia article explaining the properties of a system of linear equations.
share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.