I want to solve a linear system in 3 variables Ax=b where A is a 4×3 matrix and b is a vector.

A is the matrix A=[1 1 1 ; 1 2 1 ; 1 2 3 ; 1 4 1] while b=[1 1 1 2]
The complete 4×4 matrix A|b has rank 4:

> rank([A b'])
ans = 4

so the system has no solution.

I used linsolve:

> linsolve(A,b')
ans =

   0.6071
   0.3571
  -0.1071

which is nonsense from a mathematical point of view, given that Ax=b has no solutions!

The only thing I can imagine, is that the software is providing a least-square solution for the system, that is a solution of A'Ax=A'b; in fact you get

> linsolve(A'*A,A'*b')
ans =

   0.6071
   0.3571
  -0.1071

yet it doesn’t warn me about this: it provides the solution linsolve(A,b') as if it were an actual solution, which it isn’t!

Am I doing something wrong or missing something?