If one were to randomly generate a great many 3x4 matrices whose entries are small integers (say between -30 and +30), what fraction of the matrices could be solved through Gauss Jordan elimination?

More practically, If I were to randomly write down 3 equations in 3 variables, what is the likelihood that they can be solved uniquely?

I have been doing a lot of work with circuit analysis, and it occurred to me that this could be a good way to figure out if I set up the equations correctly. It seems to me that most sets of equations would not have a unique solution, so if I find that the system of equations DOES have a unique solution, it is unlikely that the entries are wrong.

Thanks!

Wait. What? Are you trying to fight me? You and your fancy words!

Sorry, I can't help. I don't speak smart. My math might not be great, but I do know that there are 3 types of people in this world. Those who can count, and those who can't.

So 3 linear equations, 3 unknowns. Unless you get a situation where one row is a linear combo of the other two, it will always be solvable.  

Might be better to figure out probability of generating a row that is a linear combination of other rows.  Will of course depend on probability distribution of the random variables.

So my premise is wrong? Most sets of linear equations are solvable?

Define "solvable."

A system determinant that is not zero is required.

You also need sets of three 'right hand sides' to define a system of three unknowns in three variables.

It would appear as three rows and four columns.

If any rows end up being multiples of another row the system collapses and has infinite solutions.

A plane and not the intersection of three lines IIRC.

A matrix equation has no solution if at least one of the eigenvalues of the coefficient matrix is zero. This implies a zero determinant and thus non-invertibility of the coefficient matrix.