The problem involves proving the invertibility of the sum of two invertible matrices, A and B, under the condition that the sum of their inverses, A^-1 + B^-1, is also invertible. This falls within the subject area of linear algebra and matrix theory.
Some participants have shared their attempts at manipulation and expressed frustration in finding a solution. A hint has been offered to consider the product of the matrices, which has prompted further reflection on relevant theorems regarding invertibility.
Participants note the challenge of deriving the desired result from the given conditions and equations, indicating a potential gap in information or understanding of the relationships between the matrices involved.
sweetiepi said: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.