Is the product of two independent random matrix full rank?

Question

The elements in matrix $A$ and $B$ are independent and absolute continous random variable. Is $AB$ be full rank with probability one?

First, $A$ and $B$ must be full rank with probability one.

I try to prove it with the following two ways.

1. I claim $|ABB^HA^H|$ is a continuous random variable (I don't know how to prove it.), then $P( AB \text{ is not full rank})=P(|ABB^HA^H|=0)=0$. Therefore, $AB$ is full rank almost surely.

2. Since $P(X\in Null(A) \cap Range(B) )=P(X\in Null(A) )\cdot P(X\in Range(B) )=0\cdot \{0,1\}=0$ (owing to the fact that linear subspace has Lebesgue measure 0), $AB$ is full rank almost surely.