Where the determinant came from? Behind the scenes the determinant came from noticing that if

is an endomorphism on some

-dimensional

-space

and

is some alternating

-linear form then the function

is an alternating

-linear form. But, we know that the space of alternating

-linear forms has dimension

. In particular

for some constant

. We then define

to be the

*determinant* of

. It's useful because it gives quantatative measure of invertibility. Namely,

if and only if

(or more generally if dealing with commutative rings if

is a unit). Does that at least give you start?