Linear algebra

  • Thread starter oddiseas
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  • #1
oddiseas
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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.
 

Answers and Replies

  • #2
tiny-tim
Science Advisor
Homework Helper
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Hi oddiseas! :smile:

(try using the X2 tag just above the Reply box :wink:)
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 :wink:, 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? :smile:
 
  • #3
oddiseas
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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.
 
  • #4
Dick
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Homework Helper
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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)?
 
  • #5
oddiseas
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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.
 

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