# When will constraints in an LP be satisfied as equalities?

Say $$\mathbf{X}$$ is a $$n\times m$$ matrix with no negative entries. Assume further that every row and every column have at least a non zero element. Denote $$\langle\mathbf{X}\rangle$$ the cone spanned by $$\mathbf{X}$$. Let $$\mathbf{q},\mathbf{k}, 1$$, and $$0$$ be vectors of the suitable dimension. In the following LP $$\begin{array}{ll} \text{maximize} &\quad \mathbf{q}\cdot 1\\ \text{subject to} & \quad \mathbf{X}^{\top}\mathbf{q}\leq\mathbf{k}\\ &\quad\mathbf{q}\geq 0, \end{array}$$

let $$\mathbf{q}^*$$ be its solution. I see that $$\mathbf{k}\not\in\langle\mathbf{X}\rangle\implies \mathbf{X}^{\top}\mathbf{q}^*\leq\mathbf{k}$$ (with $$\mathbf{X}^{\top}\mathbf{q}^*\ne\mathbf{k}$$), because $$\mathbf{k}\not\in\langle\mathbf{X}\rangle$$ implies that there does not exist a set of positive weights satisfying $$\mathbf{X}^{\top}\mathbf{q}=\mathbf{k}$$.

However, I believe that $$\mathbf{k}\in\langle\mathbf{X}\rangle\implies \mathbf{X}^{\top}\mathbf{q}^*=\mathbf{k}$$. Is this true? Why?

No, it is not true. Consider $$X^{T}=\left[\begin{array}{cc} 1 & 6\\ 1 & 3 \end{array}\right],\ k=\left[\begin{array}{c} 7\\ 4 \end{array}\right].$$