# Nonnegative matrix exercise by Minc

I must show that:

Let $$A$$ be a nonnegative $$n\times n$$ matrix, show that $$A$$ is reducible iff there exists a proper subset $$\{e_{j_1},e_{j_2},...,e_{j_k}\}$$ of the standard basis of $$\mathbb{R^n}$$ such that $$\subset $$

I have used the fact that there exists a permutation matrix $$P$$ such that $$M=\begin{bmatrix} B& C \\ 0& D \\ \end{bmatrix}=P^tAP$$

also $$P=[e_{j_1},...,e_{j_n}]$$ where those are column vectors and the indexes represent a permutation of the ones on the regular basis. $$AP=[Ae_{j_1},...,Ae_{j_n}]$$ and I defined the span as follows $$=\{x\in\mathbb{R^n}/x=b_1Ae_{j_1}+b_2Ae_{j_2}+...+b_kAe_{j_k}=A(b_1e_{j_1}+b_2e_{j_2}+...+b_ke_{j_k})=Ay, y\in \}$$. But I dont know how to prove this.