TITLE: Determinant map is homomorphism and surjective.
QUESTION [5 upvotes]: I just came from a course of abstract algebra, and my teacher told us that the determinant map $\det : GL(n, \mathbb{R}) \to \mathbb{R}^\times$ is a surjective homomorphism. 
Here, $GL(n, \mathbb{R}) = $ the set of $(n \times n)$ matrices $M$ such that $\det(M) \neq 0$
Why is $\det$ surjective?

REPLY [8 votes]: Note that $$\det\begin{pmatrix}a&0&\cdots & 0\\
0&1&\cdots & 0\\
\vdots&\vdots&\ddots&\vdots\\0 & 0 & \cdots &1\end{pmatrix} =a$$