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Invertible matrices problem

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  • #1
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Homework Statement


For the invertible matrices A, B and A-B, simplify the expression [tex](A - B)^{-1}A(A^{-1} - B^{-1})B[/tex].


Homework Equations


properties of invertible matrices


The Attempt at a Solution


[tex](A - B)^{-1}A(A^{-1} - B^{-1})B[/tex]
= [tex](A - B)^{-1}(AA^{-1}B - AB^{-1}B)[/tex]
= [tex](A - B)^{-1}(IB - AI)[/tex]
= [tex](A - B)^{-1}(B - A)[/tex]
= [tex]I[/tex]

Am I correct?
 

Answers and Replies

  • #2
vela
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Almost. The last step isn't quite correct.
 
  • #3
LCKurtz
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Verrrry close. Check your last step.
 
  • #4
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So it just ends here?

[tex](A - B)^{-1}(B - A)[/tex]
 
  • #5
vela
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No, you can simplify it. Consider this. Let C=B-A. Then C-1=(B-A)-1, but in your expression, you have (A-B)-1, which isn't the same matrix.
 
  • #6
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No, you can simplify it. Consider this. Let C=B-A. Then C-1=(B-A)-1, but in your expression, you have (A-B)-1, which isn't the same matrix.
Ok, so then the answer is:

[tex]-I[/tex] ?
 
  • #7
LCKurtz
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:approve:
 
  • #8
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Thanks
 

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