Invertibility of a square matrix

Question

Take a square matrix $A$ such that $$ A=BDC, $$ where $B,D,C$ are square matrices with all positive entrances; $D$ is a diagonal matrix with all entrances in the main diagonal positive.

Is it true that: if $A$ is invertible, then $B$ is invertible? Why?

Answer 1

$$\det(A)=\det(B)\det(D)\det(C)\ne0$$ and obviously $$\det(B)\ne0.$$

Answer 2

It doesn't matter that the entries are positive or that $D$ is a diagonal matrix. From $$A=BDC,$$ multiply on the right by $A^{-1}$ to get $$I=BDCA^{-1}=B\left(DCA^{-1}\right).$$ In other words, $DCA^{-1}$ is the inverse of $B$.