# How to express the 'general solution' of a matrix with a unique solution [closed]

Hi guys so I've been given some uni homework for the subject Calculus and Linear Algebra. Basically it's just your standard 'give the general solution of the system, or show that no solutions exists'. After reducing to row echelon form, I noted that the system was indeed consistent (therefore there would be a solution) and that the solution was unique - as each column contained a pivot. So through back-substitution I easily obtained values for my three variables. But this solution is a unique one - so how would I write it generally?

Doesn't 'generally' imply that there are constants contained within the solution that can be given arbitrary values (given they satisfy any restrictions) ?

Yes, I know how to write solutions very nicely when they contain free variables, using the particular solution and the nullspace... but the wording of my homework question leads me to believe that there should be some 'general' way to write a unique solution... which is kind of contradictory to the notion of 'unique'.
I've looked online and most people just leave the solution as x=2, y=4, z=3 or leave it in the variable matrix. However, this university subject is really quite strict. They expect formal solutions using mathematical symbols like ∈ and ℝ and {} brackets to denote a set (Which is intuitive when you are writing solutions with free variables). If anyone has any ideas on how I should write my unique solution, please let me know! Thanks !

Note: My solution is x=2, y=4, z=3. The matrix was rectangular, but it was 4x3 and resulted in a row of zeros when I reduced the matrix. Even so, there were no free variables because each column still contained a pivot. With my current knowledge, I am certain this won't make a difference in writing the solution out - I just thought I should mention it in case it has some significance of which I was not aware.

P.S: Sorry if the question was stupid, I'm still in high school - I only do one subject (this subject) via university

## closed as unclear what you're asking by Alex M., zz20s, Shailesh, user91500, John BApr 18 '16 at 8:51

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• It isn't "the general solution of a matrix" that you are asking about, is it? I don't know what such a thing might mean. Solutions make sense for equations, maybe a matrix equation or a linear system expressed as a matrix equation, but not "solution of a matrix". – hardmath Apr 18 '16 at 2:46

If the solution is unique, the general solution is just the solution, i.e. $(x,y,z) = (2,4,3).$ I believe the wording of the question was just in case the solution was not unique.