How to prove surjectivity of determinant It may seem obvious that the determinant of a square matrix as a map is surjective (since there is always a choice of matrix entries that yields a real number). I can't prove this statement. Any clue please?

Multiplying a row (or column) of a matrix by a constant results in multiplying the determinant by that constant. (The determinant is a $k$-multilinear function.)
So, given $r\in\Bbb R$, just take a matrix with nonzero determinant, $d$,  and multiply a row by the constant $\frac rd$, to get a matrix with determinant $r$. 
