Linear algebra Given that A B and AB are invertible matrices

Linear algebra. Given that A, B and A+B are invertible matrices. How to show that A(A^{-1}+B^{-1})B(A+B)^{-1}=1 ?

Solution

GIVEN STATEMENT

A(A^-1+B^-1)B(A+B)^-1

And we know that

XX^-1=I

so

A(A^-1+B^-1)B(A+B)^-1

=(AA^-1+AB^-1)B(A+B)^-1   (MULTPLYING A WITH (A^-1+B^-1))

=(I+AB^-1)B(A+B)^-1 (SINCE AA^-1=I)

=(B+AB^-1B)(A+B)^-1   (MULTPLYING B WITH (A^-1+B^-1))

=(B+A)(A+B)^-1   (SINCE BB^-1=I)

=(A+B)(A+B)^-1 (SINCE A+B=B+A)

=I

HENCE PROVED