Proving Invertibility of Matrix Sum: A+B

[sweetiepi]

Homework Statement

Let A and B be invertible matrices such that A^-1 + B^-1 is also invertible. Prove that A+B is invertible.

Homework Equations

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

The Attempt at a Solution

I've been trying to manipulate these equations to make something work, but I just can't seem to find the right combination.

 

[calorimetry]

Haha, me neither. I think we take the same class at the same school XD

 

[Scigatt]

Homework Statement

Let A and B be invertible matrices such that A^-1 + B^-1 is also invertible. Prove that A+B is invertible.

Homework Equations

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

The Attempt at a Solution

I've been trying to manipulate these equations to make something work, but I just can't seem to find the right combination.

Hint:Try to get A+B by multiplying the terms implied in the problem statement.

 

[calorimetry]

Thanks Scigatt. Your hint reminds me of a theorem that said "The product of invertible matrices is invertible." It's so simple when you put it like that.