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 thedeterminantof . 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?