How Do I Verify Identity and Closure in a Matrix Subgroup?

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In summary, the conversation is about a question regarding a set of matrices and how to show that the identity matrix belongs in the set and that there is closure within the set. The participants discuss the conditions for the identity matrix to be in the set and how to show closure by choosing specific values for the matrices. They also discuss the correct format for the identity matrix and clarify the rules that it must satisfy to be in the set. One participant corrects another's evaluation on closure and explains the importance of the determinant in determining if a matrix is in the set.
  • #1
elle
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Subgroup help please! Urgent!

Hi,
Can anyone help me with the following question? Many thanks! Again I'm working from a book so I'm must trying to apply theory to exercises -__-

http://i12.tinypic.com/3zsyuf8.jpg

How do I show that the identity lies in this? And also show that we have closure? Do I just choose any values of a, b and c and multiply two matrices? :confused:
 
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  • #2
elle,

What conditions do you need on a, b, and c in order to make the identity? Are they within the constraints of this set? I think so. Just choose those values of a,b,c.

In order to show inverses exist for every element of the set, you know something regarding determinants that guarantees inverses.

Choose two exeplary elements from the matrix, ie a matrix with values a,b,c and another with d,e,f. Multiply them, and see if you get another matrix of the same form.
 
  • #3
Okay, sorry I can't put this in math format but hopefully you understand what I mean :rolleyes:

In order for it to fulfill identity, I've put in ac not to be equal to 0 if a and c are not equal to 0..is that right?

And if say i take the given matrix in the question and another matrix which has elements d, e and f, lie in K, then when multiplied you get the products ad, ae+bf, cf in the new matrix? And ad, ae+bf and cf are not equal to 0 hence we have closure? :confused:
 
  • #4
What is the identity matrix? Does it satisfy the rule that tells you which matrices belong to that set?
 
  • #5
Right I can't type the identity matrix up in its correct format but its [ 1, 0, (first row) 0,1 (second row) ]. So it does satisfy the rule? :confused:

Is my evaluation on closure incorrect?
 
  • #6
Why are you asking a question about the identity. What are the two rules it, the identity, must satisfy to be in the set. Are they both satisfied? Yes.

You composition one is not correct. You say that since ad, cf and ae+bf are not zero that it is in the group. What the upper right hand entry is is immaterial. All that matters is that the lower left is zero, and the the determinant (which is ac, for the matrix in post 1) is not zero.
 
  • #7
The identity is
[tex]\left(\begin{array}{cc}1 && 0\\0 && 1\end{array}\right)[/tex]
Your general matrix is
[tex]\left(\begin{array}{cc}a && b\\0 && c\end{array}\right)[/tex]
with a and c not 0. What should a, b, c be to give the identity matrix?
 

1. What is subgroup closure?

Subgroup closure refers to the property that states that if you take any two elements from a subgroup and combine them using the group operation, the result will also be an element of that subgroup.

2. Why is subgroup closure important?

Subgroup closure is important because it ensures that the group operation is consistent and that the group remains closed under the operation. This allows for the group to maintain its structure and for calculations to be performed accurately.

3. How do you test for subgroup closure?

To test for subgroup closure, you can take any two elements from the subgroup and perform the group operation on them. If the result is also an element of the subgroup, then the subgroup is closed under the operation.

4. What is subgroup identity?

Subgroup identity refers to the element within a subgroup that when combined with any other element from the subgroup using the group operation, will result in that other element. In other words, it is the identity element for the subgroup.

5. How do you find the subgroup identity?

The subgroup identity can be found by looking for the element within the subgroup that when combined with any other element using the group operation, will result in that other element. This element is known as the identity element for the subgroup.

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