Proving Inverses (Matrix Multiplications)

[jwb]

Let A and B be n x n matrices such that I+AB is invertible.

If B is invertible what can you deduce about the matrix I+BA.

I know that I+BA is invertible but not sure how to go about proving this.

 

[micromass]

What can you tell about the matrix (I+BA)B?

 

[jwb]

it is also invertible due to being a multiplication or addition of only elementary matrices. Its not the theory I am having trouble with its the mathematical steps needed to prove that it is invertible

 

[AlephZero]

it is also invertible due to being a multiplication or addition of only elementary matrices.

I think you are gettng confused between what you "know" (i.e. you know or you have guessed what the answer is) and what you are trying to prove.

(I+BA)B = B + BAB = B(I + AB)

Now think about which of those matrices you KNOW are invertible, and which you want to PROVE are invertible.