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Let $A$ , $B$ be complex $n \times n$ matrices. Which of the followings are true?

  1. If $ A$,$B$ and $A+B$ are invertible, then $A^{-1}+B^{-1}$ is invertible.
  2. If $ A$,$B$ and $A+B$ are invertible, then $A^{-1}-B^{-1}$ is invertible.
  3. If $AB$ is nilpotent, then $BA$ is nilpotent.
  4. Characteristic polynomial of $AB$ and $BA$ are equal if $A$ is invertible.

Clearly 1 is true and I found a example in 2 that 2 is not correct, but I have no idea on 3 and 4. Kindly help me.

  • 1
    Your reasoning seems to be wrong, as $(A+B)^{-1}\neq A^{-1}+B^{-1}$ in general. Statement 1 is true nonetheless.2012-12-13

2 Answers 2

1

As someone has given an excellent hint on 3, I will give you hints on 1 and 4:

1) What is $B^{-1}(A+B)A^{-1}$?

4) Note that $\det X\det Y = \det XY = \det Y\det X$. Now try to factor something out from $\det(\lambda I - AB)$.

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    It's already clear to an extent that it almost becomes a spoiler. You could try to spend more time on it. Good luck! ;-D2012-12-13
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$BA$ is nilpotent only when $AB$ and $BA$ are commutatative. Not in general.. therefore 3 is not true..

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    You mean when $A$ and $B$ commute with each other, right? Also, you can consider adding an explicit example or a hint for such an example.2012-12-16