Isomorphism: matrix determinant

[kala]

Determine whether the given map \varphi is an isomorphism of the first binary structure with the second.
< M₂(R ), usual multiplication > with <R, usual multiplication> where \varphi(A) is the determinant of matrix A.

The determinant of the matrix is ad-bc, so \varphi(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?

 

[Hurkyl]

I'm having trouble getting this to work.

Maybe it can't...

 

[kala]

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.

 

[HallsofIvy]

You just said that you must prove the function is "one to one". That is that two different matrices, such as
\begin{bmatrix}2 &amp; 1 \\ 1 &amp; 1\end{bmatrix}
and
\begin{bmatrix}3 &amp; 2\\ 1 &amp; 1\end{bmatrix}
must not have the same determinant. Is that true?

I think you should to reread that problem.

 

[Hurkyl]

the question says it is.

No it doesn't. The question is asking you whether or not it is.

 

[kala]

Oh duh... That was stupid... It doesn't have to be. Thanks