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## Homework Statement

a)Let A be an n*n invertible matrix. Show that the inverse of the transpose of a matrix (A^{T}) is (A⁻¹)^{T}

b)Let A,B, 2A+B be n×n matrices.Show that A⁻¹+2B⁻¹ is also invertible and express (A⁻¹+2B⁻¹)⁻¹ in terms of A, B and (2A+B)⁻¹

## Homework Equations

I have done part a. I have been stuck on part b for a while. I have tried everything i can think of.

## The Attempt at a Solution

I have tried revamping

(A⁻¹+2B⁻¹)⁻¹(A⁻¹+2B⁻¹)=I

(A⁻¹+2B⁻¹)⁻¹A⁻¹+ (A⁻¹+2B⁻¹)⁻¹2B⁻¹=I

(A⁻¹+2B⁻¹)⁻¹=(I-(A⁻¹+2B⁻¹)⁻¹2B⁻¹)A

Then simplified it, but the result is not in terms of the matrix (2A+B)⁻¹,A and B.

This question really has me stumped.