Determinant from pivots

Let $A$ be a matrix that is decomposed through elimination to $A=LU$. The truth of the following statements allow us to read the determinant of $A$ from the pivots that are placed along the diagonal of $U$:

1. If $B=CD$ then $\text{det}(B)=\text{det}(C)\text{det}(D)$.
2. [what statement?]
3. All entries along the diagonal of $L$ are 1.

From these statements, we can see that the determinant of $A$ is the product of the pivots that are placed along the diagonal of $U$.