Isomorphism: matrix determinant

In summary, the given map \varphi is an isomorphism if it is one-to-one, onto, and preserves operations, where \varphi(A) is the determinant of matrix A. The function must prove that two different matrices have different determinants in order to be one-to-one. The question is asking whether or not the function is an isomorphism.
  • #1
kala
21
0
Determine whether the given map [tex]\varphi[/tex] is an isomorphism of the first binary structure with the second.
< M2(R ), usual multiplication > with <R, usual multiplication> where [tex]\varphi[/tex](A) is the determinant of matrix A.

The determinant of the matrix is ad-bc, so [tex]\varphi[/tex](A)=ad-bc.
For this to be an isomorphism, I have to show that the function is one to one, onto and preserves the operations.
I'm having trouble getting this to work. Any suggestions?
 
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  • #2
kala said:
I'm having trouble getting this to work.
Maybe it can't...
 
  • #3
According to the book it is suppose to be an isomorphism, the question says it is. I can get it to be one to one and onto, but i am having trouble with it preserving the operations.
 
  • #4
You just said that you must prove the function is "one to one". That is that two different matrices, such as
[tex]\begin{bmatrix}2 & 1 \\ 1 & 1\end{bmatrix}[/tex]
and
[tex]\begin{bmatrix}3 & 2\\ 1 & 1\end{bmatrix}[/tex]
must not have the same determinant. Is that true?

I think you should to reread that problem.
 
  • #5
kala said:
the question says it is.
No it doesn't. The question is asking you whether or not it is.
 
  • #6
Oh duh... That was stupid... It doesn't have to be. Thanks
 

FAQ: Isomorphism: matrix determinant

1. What is isomorphism?

Isomorphism is a mathematical concept that describes a one-to-one correspondence between two mathematical structures or objects. In other words, it is a type of mapping that preserves the structure and relationships of the objects being mapped.

2. What is a matrix determinant?

A matrix determinant is a scalar value that can be calculated from a square matrix. It represents the scaling factor of the corresponding linear transformation represented by the matrix.

3. How is isomorphism related to matrix determinants?

In the context of linear algebra, two square matrices are isomorphic if and only if they have the same determinant. This means that if two matrices are isomorphic, they represent the same linear transformation.

4. Can two non-square matrices be isomorphic?

No, two non-square matrices cannot be isomorphic because isomorphism only applies to square matrices. This is because non-square matrices cannot represent linear transformations.

5. How can I determine if two matrices are isomorphic?

To determine if two matrices are isomorphic, you can calculate their determinants. If the determinants are equal, then the matrices are isomorphic. Additionally, you can also check if the two matrices have the same dimensions and if they have the same number of nonzero elements in each row and column.

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