Say that we have a vector field on $GL(n,R)$, given by $A \rightarrow (A,A^2)$ (I mean the matrix power here). If we try to find the flow of this vector field, I get that it should be: $X(A,t) = e^{tA^2}A$, where $e^{tA^2}$ is the matrix exponential here. However, since I feel somewhat unsure about the material, I am not sure that my approach is correct. Could someone either give some confirmation as to that this method is correct, and if so, does it follow that our vector field will be a complete one? I strongly suspect that such is the case.
If I'm wrong I'd love some hint that might put me in the right direction.