How do you find a transition matrix with polynomials?

[rotund pawn]

https://prnt.sc/wnmQYduTxkRQ

So far, I have:
c1(1,x,x^2)=2
c1(1,x,x^2)=-4x
c1(1,x,x^2)=x^2+x+1

But when I form them into a system I get:
c1=2
c2=-4
c3=1

So does that mean the transition matrix is 3x^2-4x+2?

[hallow lotus]

3x^2-4x+2 is not a matrix. I think your first approach is closer to the answer

[rotund pawn]

.solved