Differentiability of $g(t) = f(t)^-1$, where $f : I subset mathbbR to mathrmGL(mathbbR^n)$ is differentiable [closed]

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Let $Isubset mathbbR$ be an open set and $f:I to mathrmGL(mathbbR^n)$ differentiable. Show that $g:Ito mathrmGL(mathbbR^n)$, defined by $g(t)=f(t)^-1$, is differentiable and calculate $g'$.




I have tried some calculations, but I can't prove.







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closed as off-topic by Shaun, John Ma, Xander Henderson, Taroccoesbrocco, amWhy Aug 22 at 10:58


This question appears to be off-topic. The users who voted to close gave this specific reason:


  • "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – Shaun, John Ma, Xander Henderson, Taroccoesbrocco, amWhy
If this question can be reworded to fit the rules in the help center, please edit the question.








  • 1




    What did you try?
    – Shaun
    Aug 21 at 21:26










  • @Shaun I have written g(t+h)-g(t) and tried to related with f. :(
    – 7697
    Aug 21 at 21:34










  • Do you know how to write down $A^-1$ in terms of $A$?
    – John Ma
    Aug 21 at 21:42














up vote
-1
down vote

favorite
1













Let $Isubset mathbbR$ be an open set and $f:I to mathrmGL(mathbbR^n)$ differentiable. Show that $g:Ito mathrmGL(mathbbR^n)$, defined by $g(t)=f(t)^-1$, is differentiable and calculate $g'$.




I have tried some calculations, but I can't prove.







share|cite|improve this question














closed as off-topic by Shaun, John Ma, Xander Henderson, Taroccoesbrocco, amWhy Aug 22 at 10:58


This question appears to be off-topic. The users who voted to close gave this specific reason:


  • "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – Shaun, John Ma, Xander Henderson, Taroccoesbrocco, amWhy
If this question can be reworded to fit the rules in the help center, please edit the question.








  • 1




    What did you try?
    – Shaun
    Aug 21 at 21:26










  • @Shaun I have written g(t+h)-g(t) and tried to related with f. :(
    – 7697
    Aug 21 at 21:34










  • Do you know how to write down $A^-1$ in terms of $A$?
    – John Ma
    Aug 21 at 21:42












up vote
-1
down vote

favorite
1









up vote
-1
down vote

favorite
1






1






Let $Isubset mathbbR$ be an open set and $f:I to mathrmGL(mathbbR^n)$ differentiable. Show that $g:Ito mathrmGL(mathbbR^n)$, defined by $g(t)=f(t)^-1$, is differentiable and calculate $g'$.




I have tried some calculations, but I can't prove.







share|cite|improve this question















Let $Isubset mathbbR$ be an open set and $f:I to mathrmGL(mathbbR^n)$ differentiable. Show that $g:Ito mathrmGL(mathbbR^n)$, defined by $g(t)=f(t)^-1$, is differentiable and calculate $g'$.




I have tried some calculations, but I can't prove.









share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Aug 22 at 9:53









Brahadeesh

4,13631550




4,13631550










asked Aug 21 at 21:14









7697

2489




2489




closed as off-topic by Shaun, John Ma, Xander Henderson, Taroccoesbrocco, amWhy Aug 22 at 10:58


This question appears to be off-topic. The users who voted to close gave this specific reason:


  • "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – Shaun, John Ma, Xander Henderson, Taroccoesbrocco, amWhy
If this question can be reworded to fit the rules in the help center, please edit the question.




closed as off-topic by Shaun, John Ma, Xander Henderson, Taroccoesbrocco, amWhy Aug 22 at 10:58


This question appears to be off-topic. The users who voted to close gave this specific reason:


  • "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – Shaun, John Ma, Xander Henderson, Taroccoesbrocco, amWhy
If this question can be reworded to fit the rules in the help center, please edit the question.







  • 1




    What did you try?
    – Shaun
    Aug 21 at 21:26










  • @Shaun I have written g(t+h)-g(t) and tried to related with f. :(
    – 7697
    Aug 21 at 21:34










  • Do you know how to write down $A^-1$ in terms of $A$?
    – John Ma
    Aug 21 at 21:42












  • 1




    What did you try?
    – Shaun
    Aug 21 at 21:26










  • @Shaun I have written g(t+h)-g(t) and tried to related with f. :(
    – 7697
    Aug 21 at 21:34










  • Do you know how to write down $A^-1$ in terms of $A$?
    – John Ma
    Aug 21 at 21:42







1




1




What did you try?
– Shaun
Aug 21 at 21:26




What did you try?
– Shaun
Aug 21 at 21:26












@Shaun I have written g(t+h)-g(t) and tried to related with f. :(
– 7697
Aug 21 at 21:34




@Shaun I have written g(t+h)-g(t) and tried to related with f. :(
– 7697
Aug 21 at 21:34












Do you know how to write down $A^-1$ in terms of $A$?
– John Ma
Aug 21 at 21:42




Do you know how to write down $A^-1$ in terms of $A$?
– John Ma
Aug 21 at 21:42










1 Answer
1






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up vote
1
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accepted










$g(t) = f(t)^-1$ is differentiable because $f(t)$ is differentiable and each entry of $f(t)^-1$ is a rational function of the entries of $f(t)$.



To compute the derivative of $g$, use $mathrmId = f(t) g(t)$ for all $t in I$ and then apply the chain rule.






share|cite|improve this answer



























    1 Answer
    1






    active

    oldest

    votes








    1 Answer
    1






    active

    oldest

    votes









    active

    oldest

    votes






    active

    oldest

    votes








    up vote
    1
    down vote



    accepted










    $g(t) = f(t)^-1$ is differentiable because $f(t)$ is differentiable and each entry of $f(t)^-1$ is a rational function of the entries of $f(t)$.



    To compute the derivative of $g$, use $mathrmId = f(t) g(t)$ for all $t in I$ and then apply the chain rule.






    share|cite|improve this answer
























      up vote
      1
      down vote



      accepted










      $g(t) = f(t)^-1$ is differentiable because $f(t)$ is differentiable and each entry of $f(t)^-1$ is a rational function of the entries of $f(t)$.



      To compute the derivative of $g$, use $mathrmId = f(t) g(t)$ for all $t in I$ and then apply the chain rule.






      share|cite|improve this answer






















        up vote
        1
        down vote



        accepted







        up vote
        1
        down vote



        accepted






        $g(t) = f(t)^-1$ is differentiable because $f(t)$ is differentiable and each entry of $f(t)^-1$ is a rational function of the entries of $f(t)$.



        To compute the derivative of $g$, use $mathrmId = f(t) g(t)$ for all $t in I$ and then apply the chain rule.






        share|cite|improve this answer












        $g(t) = f(t)^-1$ is differentiable because $f(t)$ is differentiable and each entry of $f(t)^-1$ is a rational function of the entries of $f(t)$.



        To compute the derivative of $g$, use $mathrmId = f(t) g(t)$ for all $t in I$ and then apply the chain rule.







        share|cite|improve this answer












        share|cite|improve this answer



        share|cite|improve this answer










        answered Aug 22 at 9:50









        Brahadeesh

        4,13631550




        4,13631550












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