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?

Hint:
Given  $r\in \Bbb R $, take the  diagonal matrix 
$\begin{bmatrix}r&0&\cdots & 0\\
0&1&\cdots & 0\\
\vdots&\vdots&\ddots&\vdots\\0 & 0 & \cdots &1\end{bmatrix} $.
