Mixing
Show set of all n x n matrices with det = 1 forms a subgroup of a general linear group
$begingroup$
https://i.gyazo.com/a343ea5c375f88f36aeea0ae31a76efe.png
In the question above, I am stuck on how to proceed. I know that the formula that I need to use at some point is $det(AB) = det(A)det(B)$.
Thanks for any help.
matrices group-theory determinant
asked Oct 13 '17 at 8:26
kriskrosskriskross
123
$endgroup$
add a comment |
$begingroup$
https://i.gyazo.com/a343ea5c375f88f36aeea0ae31a76efe.png
In the question above, I am stuck on how to proceed. I know that the formula that I need to use at some point is $det(AB) = det(A)det(B)$.
Thanks for any help.
matrices group-theory determinant
asked Oct 13 '17 at 8:26
kriskrosskriskross
123
$endgroup$
add a comment |
$begingroup$
https://i.gyazo.com/a343ea5c375f88f36aeea0ae31a76efe.png
In the question above, I am stuck on how to proceed. I know that the formula that I need to use at some point is $det(AB) = det(A)det(B)$.
Thanks for any help.
matrices group-theory determinant
asked Oct 13 '17 at 8:26
kriskrosskriskross
123
$endgroup$
https://i.gyazo.com/a343ea5c375f88f36aeea0ae31a76efe.png
In the question above, I am stuck on how to proceed. I know that the formula that I need to use at some point is $det(AB) = det(A)det(B)$.
Thanks for any help.
matrices group-theory determinant
matrices group-theory determinant
asked Oct 13 '17 at 8:26
kriskrosskriskross
123
asked Oct 13 '17 at 8:26
kriskrosskriskross
123
asked Oct 13 '17 at 8:26
kriskrosskriskross
123
asked Oct 13 '17 at 8:26
kriskrosskriskross
123
asked Oct 13 '17 at 8:26
kriskrosskriskross
123
123
add a comment |
add a comment |
2 Answers
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$begingroup$
Let $G$ be the set of all $n times n$ matrices witt $det =1$.
You have to show:
If $A,B in G$, then $AB in G$ and $A^{-1} in G$.
If $A,B in G$, then $det(AB)=x$ and $det(A^{-1})=y$.
If you can show that $x=y=1$, then you are done !
answered Oct 13 '17 at 8:32
FredFred
48.8k11849
$endgroup$
$begingroup$
Ah thanks! What should I use for A and B?
$endgroup$
– kriskross
Oct 13 '17 at 8:55
add a comment |
$begingroup$
If $G$ is a group and $H$ is a subset of $G$, then $H$ is a subgroup if and only if
- For each $h_1,h_2in H$, the element $h_1h_2$ is also in $H$
- For each $hin H$, the element $h^{-1}$ is also in $H$.
That's where you start. At the definition.
You can also use the fact that in your case, an element $A$ (a matrix) is in $H$ if and only if $det(A)=1$.
edited Jan 28 at 13:02
user639631
506
answered Oct 13 '17 at 8:34
5xum5xum
91.8k394161
$endgroup$
add a comment |
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2 Answers
2
active
oldest
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2 Answers
2
active
oldest
votes
active
oldest
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active
oldest
votes
$begingroup$
Let $G$ be the set of all $n times n$ matrices witt $det =1$.
You have to show:
If $A,B in G$, then $AB in G$ and $A^{-1} in G$.
If $A,B in G$, then $det(AB)=x$ and $det(A^{-1})=y$.
If you can show that $x=y=1$, then you are done !
answered Oct 13 '17 at 8:32
FredFred
48.8k11849
$endgroup$
$begingroup$
Ah thanks! What should I use for A and B?
$endgroup$
– kriskross
Oct 13 '17 at 8:55
add a comment |
$begingroup$
Let $G$ be the set of all $n times n$ matrices witt $det =1$.
You have to show:
If $A,B in G$, then $AB in G$ and $A^{-1} in G$.
If $A,B in G$, then $det(AB)=x$ and $det(A^{-1})=y$.
If you can show that $x=y=1$, then you are done !
answered Oct 13 '17 at 8:32
FredFred
48.8k11849
$endgroup$
$begingroup$
Ah thanks! What should I use for A and B?
$endgroup$
– kriskross
Oct 13 '17 at 8:55
add a comment |
$begingroup$
Let $G$ be the set of all $n times n$ matrices witt $det =1$.
You have to show:
If $A,B in G$, then $AB in G$ and $A^{-1} in G$.
If $A,B in G$, then $det(AB)=x$ and $det(A^{-1})=y$.
If you can show that $x=y=1$, then you are done !
answered Oct 13 '17 at 8:32
FredFred
48.8k11849
$endgroup$
Let $G$ be the set of all $n times n$ matrices witt $det =1$.
You have to show:
If $A,B in G$, then $AB in G$ and $A^{-1} in G$.
If $A,B in G$, then $det(AB)=x$ and $det(A^{-1})=y$.
If you can show that $x=y=1$, then you are done !
answered Oct 13 '17 at 8:32
FredFred
48.8k11849
answered Oct 13 '17 at 8:32
FredFred
48.8k11849
answered Oct 13 '17 at 8:32
FredFred
48.8k11849
answered Oct 13 '17 at 8:32
FredFred
48.8k11849
48.8k11849
$begingroup$
Ah thanks! What should I use for A and B?
$endgroup$
– kriskross
Oct 13 '17 at 8:55
add a comment |
$begingroup$
Ah thanks! What should I use for A and B?
$endgroup$
– kriskross
Oct 13 '17 at 8:55
$begingroup$
Ah thanks! What should I use for A and B?
$endgroup$
– kriskross
Oct 13 '17 at 8:55
$begingroup$
Ah thanks! What should I use for A and B?
$endgroup$
– kriskross
Oct 13 '17 at 8:55
add a comment |
$begingroup$
If $G$ is a group and $H$ is a subset of $G$, then $H$ is a subgroup if and only if
- For each $h_1,h_2in H$, the element $h_1h_2$ is also in $H$
- For each $hin H$, the element $h^{-1}$ is also in $H$.
That's where you start. At the definition.
You can also use the fact that in your case, an element $A$ (a matrix) is in $H$ if and only if $det(A)=1$.
edited Jan 28 at 13:02
user639631
506
answered Oct 13 '17 at 8:34
5xum5xum
91.8k394161
$endgroup$
add a comment |
$begingroup$
If $G$ is a group and $H$ is a subset of $G$, then $H$ is a subgroup if and only if
- For each $h_1,h_2in H$, the element $h_1h_2$ is also in $H$
- For each $hin H$, the element $h^{-1}$ is also in $H$.
That's where you start. At the definition.
You can also use the fact that in your case, an element $A$ (a matrix) is in $H$ if and only if $det(A)=1$.
edited Jan 28 at 13:02
user639631
506
answered Oct 13 '17 at 8:34
5xum5xum
91.8k394161
$endgroup$
add a comment |
$begingroup$
If $G$ is a group and $H$ is a subset of $G$, then $H$ is a subgroup if and only if
- For each $h_1,h_2in H$, the element $h_1h_2$ is also in $H$
- For each $hin H$, the element $h^{-1}$ is also in $H$.
That's where you start. At the definition.
You can also use the fact that in your case, an element $A$ (a matrix) is in $H$ if and only if $det(A)=1$.
edited Jan 28 at 13:02
user639631
506
answered Oct 13 '17 at 8:34
5xum5xum
91.8k394161
$endgroup$
If $G$ is a group and $H$ is a subset of $G$, then $H$ is a subgroup if and only if
- For each $h_1,h_2in H$, the element $h_1h_2$ is also in $H$
- For each $hin H$, the element $h^{-1}$ is also in $H$.
That's where you start. At the definition.
You can also use the fact that in your case, an element $A$ (a matrix) is in $H$ if and only if $det(A)=1$.
edited Jan 28 at 13:02
user639631
506
answered Oct 13 '17 at 8:34
5xum5xum
91.8k394161
edited Jan 28 at 13:02
user639631
506
edited Jan 28 at 13:02
user639631
506
edited Jan 28 at 13:02
user639631
506
506
answered Oct 13 '17 at 8:34
5xum5xum
91.8k394161
answered Oct 13 '17 at 8:34
5xum5xum
91.8k394161
answered Oct 13 '17 at 8:34
5xum5xum
91.8k394161
91.8k394161
add a comment |
add a comment |
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