prove DC^-1 BA^-1=(AB^-1 CD^-1)^-1. First state what you need to show; and then show it properly.

Question

Matrices Determinants

Answer 1

You must show that DC^-1BA^-1 when multiplied by AB^-1CD^-1 yields the Identity Matrix.

 

(DC^-1BA^-1)(AB^1CD^1)=  I       to prove product of expressions = I

(DC^-1B)(A^-1A)(B^1CD^1)= I      Associative Law

(DC^-1B)(I)(B^1CD^1)= I              Multiplicative Inverses

(DC^-1B)(B^1CD^1)= I                  Multiplicative Identity 

(DC^-1)(BB^1)(CD^1)=  I              Associative Law

(DC^-1)(I)(CD^1)= I                      Multiplicative Inverses

(DC^-1)(CD^1)= I                          Multiplicative Identity

D(C^-1C)D^1= I                             Associative Law

D(I)D^-1= I                                    Multiplicative Inverses                             DD^-1 = I                                       Multiplicative Identity

I = I                                               Multiplicative Inverses