# Parametric solutions to system of equations inequations via solve()

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Arthur on 8 Nov 2014
Commented: Arthur on 13 Nov 2014
Given a square matrix A in n x n symbolic variables a_{i,j}, I want a parametric solutions for the a_{i,j} constrained to a system of equations and inequalities.
For instance, given
syms a, b, c, d, x
A = [[a, b]; [c, d]]
e(x) = charpoly(A, x)
I would like to find parametric solutions for all a, b, c, d that satisfy, for example
e(1)==0 and a >= 0, b >= 0, c >= 0, d >= 0.
Maple has functionality like this ("Solve semi-algebraic", or something) and I would like to be able to do this with Matlab.
I attempted to use solve() (see attached screenshot), but, even in this simpler case I consider there, I get the warning and the empty solution, so I'm guessing that I'm not understanding the solve() function correctly.
Thank you.

Stefan Wehmeier on 11 Nov 2014
I see that you are using R2014a. Then your only option is to solve for one variable
solve(e(1) == 0, a)
and take this as a parameterization with which you can play by plugging in values for two variables, and solving result>=0 for the third.
If you upgrade to R2014b, just
solve([e(1) == 0, a>=0, b>=0, c>=0, d>=0], [a, b, c, d])
works and gives you some solutions.
Arthur on 13 Nov 2014
Alas, there is no student edition of R2014b. Strange that this feature would be unavailable until that release.