Math Forum :: View topic – matrix

Let A be an nxn matrix,B be an nxr matrix,C be an rxn matrix and D be an rxr matrix,all over a commutative ring R with identity.Suppose there is a matrix E over R such that DE=C.Prove that det[A B]=det(A-BE)detD. C D

What conclusion can be drawn if there is a matrix F over R such that FD=B?