what are the relations between the constant (a,b and c) so that the following system of linear equations has a unique solution ? $$ \begin{cases} -x+y+2z=a\\ 3x-y+z=b\\ -x+3y+4z=c\\ \end{cases} $$
I used Gaussian elimination to reach this : $$\left(\begin{matrix}1&0&0&(-7a+2b+3c)/10\\0&1&0&(-13a-2b+7c)/10\\0&0&1&(4a+b-c)/5\end{matrix}\right)$$ and the determinant of $$\left(\begin{matrix}-1&1&2\\3&-1&1\\-1&3&4\end{matrix}\right)$$ != 0 but i don't know how to infer the relations between (a,b,c) ?
If the determinant is non-zero, it has an inverse and hence it always have a unique solution.
No constraint has to be imposed on $a,b,c$.