Thread: Invertible Matrices

Hello!

This is one problem I've been tackling for a while:

If A, B, and A+B are all invertible matrices of order "n", prove that is invertible and that:



First, is there a generic formula for ? (If so, our instructor never taught us one). I'm also assuming that we can't assume AB is commutable, because that could make everything so much easier. ^_^;

Anyway. Any helpful hints might help. Basically what I've tried to do so far is equate the "middle" and RHS statements by multiplying both sides by inverses to isolate , but I don't really see how that 'proves' anything ...

Originally Posted by ginarific

Hello!

This is one problem I've been tackling for a while:

If A, B, and A+B are all invertible matrices of order "n", prove that is invertible and that:



First, is there a generic formula for ? (If so, our instructor never taught us one). I'm also assuming that we can't assume AB is commutable, because that could make everything so much easier. ^_^;

Anyway. Any helpful hints might help. Basically what I've tried to do so far is equate the "middle" and RHS statements by multiplying both sides by inverses to isolate , but I don't really see how that 'proves' anything ...

How do you show a matrix is invertible? Find a matrix that multiplied by the first one gives you the identity one!
Well, let us show that is the inverse of the matrix :

and we're done!

Tonio