i need to prove the following:

let .

this is what i've tried so far:

( )

iff A is invertible.

since A is invertible, it can be presented as a product of elementary transformations:

and then:

now, since E is either , , or its determinant is 1, -1 or respectivly, then:

my problem is with my last sentence: i'm kinda using what i need to prove in my proof...

that's cheating.

i can really use a guidance here.

plus, how can i prove the other direction? ( )

thanks in advanced!