TITLE: Proving the product of two non singular matrices is also non singular.
QUESTION [6 upvotes]: I am having trouble with a proof for linear algebra. Could somebody explain to me how to prove that if $A$ and $B$ are both $n\times n$ non singular matrices, that their product $AB$ is also non singular. 
A place to start would be helpful. Thank you for your time.

REPLY [6 votes]: There's different manners to prove this result for example:

Using the determinant:
$$\det(AB)=\det A\det B$$
and the fact that $C$ is singular iff $\det C=0$.
Using the fact that $AB$ is invertible then $A$ is surjective and $B$ is injective and that in finite dimensional space: $C$ is injective iff $C$ is surjective iff $C$ is bijective.

REPLY [5 votes]: Note that a matrix is non-singular if and only if it has an inverse.
Suppose $A$ and $B$ have inverses $A^{-1} B^{-1}$.  What do you get when you multiply
$$
(AB)(B^{-1}A^{-1})
$$
and why can we now conclude that $AB$ is non-singular?