Given minimal polynomial and equation, determine Matrix so the following exercise:
find a real 3x3 Matrix A with $ A*\begin{pmatrix} 1 \\ 1 \\ 0 \end{pmatrix}=\begin{pmatrix} 0 \\ 1 \\ 1 \end{pmatrix}  $ and minimal polynomial $p=t^2*(t+1)$.
I know how the JCF looks here (1 Jordan block of size 2 with Eigenvalue 0 and one Jordan block of size 1 with Eigenvalue 1) and I know that the equation says something about the entries of A for instance $ a_{21}=-a_{11}$. But how do I determine such a Matrix now? Am I just supposed to trial and error for the entries of A and see if I get one with the appropriate Eigenvalues etc. or is there a better way ?
 A: Hint: Let $J$ denote the Jordan normal form of $A$, i.e.
$$
J = \pmatrix{0&1\\&0\\&&1}.
$$
We know that $J$ satisfies the following:
$$
J \pmatrix{0\\1\\0} = \pmatrix{1\\0\\0}, \quad J\pmatrix{1\\0\\0} = \pmatrix{0\\0\\0}, \quad J \pmatrix{0\\0\\1} = \pmatrix{0\\0\\1}.
$$
With that in mind, look for a change of basis that can be applied to $J$ to produce a matrix satisfying the requirements of the question. In other words, we want an invertible $S$ such that $SJS^{-1}$ satisfies the requirements of the question.
As a further hint, two columns of $S$ can be taken directly from the question statement.
A: I followed Ben Grossmann's idea. Let $J=\left(\begin{matrix} 0 & 1 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & -1 \\\end{matrix}\right)$. Then, in particular $Je_2=e_1$. We now choose an invertible matrix $S$ such that $S: e_2\rightarrow e_1+e_2, e_1\rightarrow e_2+e_3$. These asignments determine second and first rows. Hence, for example, we can let $S=\left(\begin{matrix} 0 & 1 & 1 \\ 1 & 1 & 0 \\ 1 & 0 & 0 \\\end{matrix}\right)$. Then, $A=SJS^{-1}=\left(\begin{matrix} -1 & 1 & -1 \\ 0 & 1 & -1 \\ 0 & 1 & -1 \\\end{matrix}\right)$.
