Consider the vector space $\mathbb{R}^3$ and the following system of vectors:
$$\mathbb{B} = \{(k+1,k,1-k),(-1,1,1),(0,k,k)\}, \ \text{where} \ k \in \mathbb{R}.$$Determine for which values of $k$, $B$ is a basis for $\mathbb{R}^3$.\begin{pmatrix}
k+1 & k & 1-k \\
-1 & 1 & 1 \\
0 & k & k \\
\end{pmatrix}