Let $G$ be the set of all $2 times 2$ symmetric invertible matrices with real entries [closed]

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Let $G$ be the set of all $2 times 2$ symmetric invertible matrices with real entries then with matrix multiplication, $G$ is not from a group. Help me to find a counter examples.




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closed as off-topic by Alan Wang, Gerry Myerson, Brian Borchers, Delta-u, Derek Holt Jul 28 at 8:16


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  • Duplicated. Here is the answer: math.stackexchange.com/questions/1979491/…
    – Carlos Jiménez
    Jul 28 at 1:01










  • I noticed that you had asked two questions before and both of them were closed. Please write your question properly using LaTeX and show in details which part you do not understand or get stucked with to prevent your question to be closed.
    – Alan Wang
    Jul 28 at 1:57







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    Possible duplicate of Group of symmetric invertible matrices
    – Gerry Myerson
    Jul 28 at 2:52














up vote
-2
down vote

favorite












Let $G$ be the set of all $2 times 2$ symmetric invertible matrices with real entries then with matrix multiplication, $G$ is not from a group. Help me to find a counter examples.




group-theory




share|cite|improve this question













closed as off-topic by Alan Wang, Gerry Myerson, Brian Borchers, Delta-u, Derek Holt Jul 28 at 8:16


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." – Alan Wang, Brian Borchers, Derek Holt
If this question can be reworded to fit the rules in the help center, please edit the question.












  • Duplicated. Here is the answer: math.stackexchange.com/questions/1979491/…
    – Carlos Jiménez
    Jul 28 at 1:01










  • I noticed that you had asked two questions before and both of them were closed. Please write your question properly using LaTeX and show in details which part you do not understand or get stucked with to prevent your question to be closed.
    – Alan Wang
    Jul 28 at 1:57







  • 1




    Possible duplicate of Group of symmetric invertible matrices
    – Gerry Myerson
    Jul 28 at 2:52












up vote
-2
down vote

favorite









up vote
-2
down vote

favorite











Let $G$ be the set of all $2 times 2$ symmetric invertible matrices with real entries then with matrix multiplication, $G$ is not from a group. Help me to find a counter examples.




group-theory




share|cite|improve this question













Let $G$ be the set of all $2 times 2$ symmetric invertible matrices with real entries then with matrix multiplication, $G$ is not from a group. Help me to find a counter examples.





group-theory





share|cite|improve this question












share|cite|improve this question




share|cite|improve this question








edited Jul 28 at 1:25









David G. Stork

7,3332828




7,3332828









asked Jul 28 at 0:57









Nidhi yadav

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closed as off-topic by Alan Wang, Gerry Myerson, Brian Borchers, Delta-u, Derek Holt Jul 28 at 8:16


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." – Alan Wang, Brian Borchers, Derek Holt
If this question can be reworded to fit the rules in the help center, please edit the question.




closed as off-topic by Alan Wang, Gerry Myerson, Brian Borchers, Delta-u, Derek Holt Jul 28 at 8:16


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." – Alan Wang, Brian Borchers, Derek Holt
If this question can be reworded to fit the rules in the help center, please edit the question.











  • Duplicated. Here is the answer: math.stackexchange.com/questions/1979491/…
    – Carlos Jiménez
    Jul 28 at 1:01










  • I noticed that you had asked two questions before and both of them were closed. Please write your question properly using LaTeX and show in details which part you do not understand or get stucked with to prevent your question to be closed.
    – Alan Wang
    Jul 28 at 1:57







  • 1




    Possible duplicate of Group of symmetric invertible matrices
    – Gerry Myerson
    Jul 28 at 2:52
















  • Duplicated. Here is the answer: math.stackexchange.com/questions/1979491/…
    – Carlos Jiménez
    Jul 28 at 1:01










  • I noticed that you had asked two questions before and both of them were closed. Please write your question properly using LaTeX and show in details which part you do not understand or get stucked with to prevent your question to be closed.
    – Alan Wang
    Jul 28 at 1:57







  • 1




    Possible duplicate of Group of symmetric invertible matrices
    – Gerry Myerson
    Jul 28 at 2:52















Duplicated. Here is the answer: math.stackexchange.com/questions/1979491/…
– Carlos Jiménez
Jul 28 at 1:01




Duplicated. Here is the answer: math.stackexchange.com/questions/1979491/…
– Carlos Jiménez
Jul 28 at 1:01












I noticed that you had asked two questions before and both of them were closed. Please write your question properly using LaTeX and show in details which part you do not understand or get stucked with to prevent your question to be closed.
– Alan Wang
Jul 28 at 1:57





I noticed that you had asked two questions before and both of them were closed. Please write your question properly using LaTeX and show in details which part you do not understand or get stucked with to prevent your question to be closed.
– Alan Wang
Jul 28 at 1:57





1




1




Possible duplicate of Group of symmetric invertible matrices
– Gerry Myerson
Jul 28 at 2:52




Possible duplicate of Group of symmetric invertible matrices
– Gerry Myerson
Jul 28 at 2:52










1 Answer
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Take $A=beginbmatrix 1 & 2 \ 2 & 1 endbmatrix$,
$B=beginbmatrix 1 & 0 \ 0 & 2 endbmatrix$. Clearly, $A,B$ are symmetric and they are elements of $G$.

Then $AB=beginbmatrix 1 & 4 \ 2 & 2 endbmatrix$.

Since $AB$ is not symmetric, the set is not closed under multiplication and hence does not form a group.






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    1 Answer
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    1 Answer
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    up vote
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    down vote













    Take $A=beginbmatrix 1 & 2 \ 2 & 1 endbmatrix$,
    $B=beginbmatrix 1 & 0 \ 0 & 2 endbmatrix$. Clearly, $A,B$ are symmetric and they are elements of $G$.

    Then $AB=beginbmatrix 1 & 4 \ 2 & 2 endbmatrix$.

    Since $AB$ is not symmetric, the set is not closed under multiplication and hence does not form a group.






    share|cite|improve this answer

























      up vote
      2
      down vote













      Take $A=beginbmatrix 1 & 2 \ 2 & 1 endbmatrix$,
      $B=beginbmatrix 1 & 0 \ 0 & 2 endbmatrix$. Clearly, $A,B$ are symmetric and they are elements of $G$.

      Then $AB=beginbmatrix 1 & 4 \ 2 & 2 endbmatrix$.

      Since $AB$ is not symmetric, the set is not closed under multiplication and hence does not form a group.






      share|cite|improve this answer























        up vote
        2
        down vote










        up vote
        2
        down vote









        Take $A=beginbmatrix 1 & 2 \ 2 & 1 endbmatrix$,
        $B=beginbmatrix 1 & 0 \ 0 & 2 endbmatrix$. Clearly, $A,B$ are symmetric and they are elements of $G$.

        Then $AB=beginbmatrix 1 & 4 \ 2 & 2 endbmatrix$.

        Since $AB$ is not symmetric, the set is not closed under multiplication and hence does not form a group.






        share|cite|improve this answer













        Take $A=beginbmatrix 1 & 2 \ 2 & 1 endbmatrix$,
        $B=beginbmatrix 1 & 0 \ 0 & 2 endbmatrix$. Clearly, $A,B$ are symmetric and they are elements of $G$.

        Then $AB=beginbmatrix 1 & 4 \ 2 & 2 endbmatrix$.

        Since $AB$ is not symmetric, the set is not closed under multiplication and hence does not form a group.







        share|cite|improve this answer













        share|cite|improve this answer



        share|cite|improve this answer











        answered Jul 28 at 1:08









        Alan Wang

        4,126931




        4,126931












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