Solving Part b of an Invertible Matrix Homework Statement

[oddiseas]

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.

 

[tiny-tim]

Hi oddiseas!

(try using the X² tag just above the Reply box )

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
…
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.

Hint: start with the result , and see what values of p q and r make it work …

multiply (A^-1 + 2B^-1) by pA + qB + r(2A + B)^-1 …

what do you get? ​

 

[oddiseas]

I have multiplied (A⁻¹+B⁻¹) by what you have suggested and i get:

p+2q+qBA⁻¹+2pAB⁻¹+r(2A²+BA)⁻¹+r(2B²+4AB)⁻¹

But i still cannot see the logic. This question just seems to get longer and longer!
Could you explain to me the logic in how i should approach this question.

 

[Dick]

I have multiplied (A⁻¹+B⁻¹) by what you have suggested and i get:

p+2q+qBA⁻¹+2pAB⁻¹+r(2A²+BA)⁻¹+r(2B²+4AB)⁻¹

But i still cannot see the logic. This question just seems to get longer and longer!
Could you explain to me the logic in how i should approach this question.

It said (2B^(-1)+A^(-1))^(-1) is a combination of the given matrices. It didn't say it was a linear combination. You just have to fool around with combinations. Hint: what is A^(-1)*(2A+B)*B^(-1)?

 

[oddiseas]

Thanks for the reply, that helps a lot. Your'e a superstar! I will work on that and see what i can get, and then post it.