Construct a matrix with the requisite properties or explain why no such marix exists. I have one with these properties: nullspace contains [1 0 1], [-1 2 1] and the row space contains [1 1 -1]
This must be a matrix wit three columns, due to the length of the vector in the row space. I kow that we find the wectors in the nullspece by computing Rx = 0 so 
x = s[1 0 1] + t[-1 2 1], but it might contain one other vector...
How do I proceede?
I also have one with the properties the column space and the nullspace both have basis [1 0]. I think this means that the matrix has 2 rows and the first columnn in matrix should be 1 0. the second row in R (row reduced A) must also be all 0. 
 A: More precisely, the column vectors
$$
\begin{bmatrix}1\\0\\1\end{bmatrix}
\qquad\text{and}\qquad
\begin{bmatrix}-1\\2\\1\end{bmatrix}
$$
should belong to the null space of the requested matrix $R$. Since they're linearly independent, this implies $\dim N(R)\ge2$. Since the row vector $[1\;1\;{-1}]$ should belong to the row space, the rank of $R$ must be $\ge1$.
By the rank-nullity theorem, we conclude that $\dim N(R)=2$ and the rank is $1$. Therefore each row of the $3\times3$ matrix $R$ is a scalar multiple of $[1\;1\;{-1}]$.
Can you finish?
A: The row space of $A $ is the range of $A^T $, which is the orthogonal of the nullspace. The three vectors you are given are pairwise orthogonal, so no obstruction. We can take for instance  $A $ to be
$$A=\begin{bmatrix}1\\1\\-1\end{bmatrix}\begin{bmatrix}1&1&-1\end{bmatrix}=\begin{bmatrix}1&1&-1\\1&1&-1\\-1&-1&1\end{bmatrix}. $$
A: Note that the row vectors you give are all orthogonal and linearly independent, so normalize them and stack them into the rows of $A$
\begin{equation}\label{eqA1}
A
=
\begin{bmatrix}
\frac{1}{\sqrt{2}} & 0 &  \frac{1}{\sqrt{2}} \\
\frac{-1}{\sqrt{6}} & \frac{2}{\sqrt{6}} & \frac{1}{\sqrt{6}} \\
\frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} & \frac{-1}{\sqrt{3}}
\end{bmatrix}
\end{equation}
This matrix  could serve as eigenvectors of the matrix you would like to construct. Let $B$ be the matrix to  construct, then $$B = A^T \Sigma A$$
where $$\Sigma = \begin{bmatrix}
0 & 0 &  0 \\
0 & 0 & 0\\
0& 0 & \lambda
\end{bmatrix}
$$
and $\lambda \neq 0$ is of your choice.
