TITLE: Invertible Matrices and Proving The Operations Equal to Identity
QUESTION [0 upvotes]: Show that if $A$, $B$, and  $A + B$  are invertible matrices with the same size, then
$$A(A^{–1} + B^{–1})B(A + B)^{–1} = I.$$
This is what i got:
$$A^{-1}A+B^{-1}ABA^{-1}+B^{-1}B = I$$
$$I+B^{-1}ABA^{-1}+B^{-1}B = I$$
$$B^{-1}ABA^{-1}+I = I $$
$$B^{-1}ABA^{-1} = I $$
$$I = I$$
$$I$$
So i have to write in proof form.

REPLY [1 votes]: What you're doing looks tricky to me. It looks like you're assuming commutation of matrix multiplication, which doesn't generally hold. What does hold is distributivity (with respect to addition), i.e. $ A (B+C)=AB+AC $. We can use this to find:
$$ A (A^{-1}+B^{-1}) B (A+B)^{-1}=(AA^{-1} B+AB^{-1} B)(A+B)^{-1} =(B+A)(A+B)^{-1}$$
And now, since matrix addition ís commutative:
$$=(A+B)(A+B)^{-1}=I $$
{"set_name": "stack_exchange", "score": 0, "question_id": 1170745}