Solution

The matrix must have the vectors and as a basis for its null space.

If we take B to be the matrix which has the vectors and for rows then we can

find a basis for its nullspace and make these the rows of a matrix. Depending on how you calculate this basis you may get different bases. One basis is the rows of the matrix

This matrix has rank 2 so its nullspace has dimension 2. The rows of B are independent and in the nullspace of A so they are a basis for the nullspace of A. This means that the general solution to Ax = is any particular solution to

plus a general element of the nullspace. That is + . If this is a solution to Ax=b then the b must be so b = =

>

is simply the null space of the transpose of the matrix obtained by augmenting and taking transpose. Clearly, the vectors will be in its null space, by definition. The rank of the matrix is clearly 2 since the chosen vectors are independent. (They clearly have a 2 by 2 nonzero subdeterminant chosen from the second and the third rows.)
Since has rank 2, its null space will have dimension 2 (or number of columns - rank). So, our vectors must be a basis of the column space. If we simply take to be , we have all we need!