A nonsingular matrix written in the form LU.

Question

Here is the paragraph from Golan's "Linear Algebra":

But I do not understand why U must be nonsingular, could anyone explains this for me please?

Answer 1

If $U$ were singular then $LU$ were singular as well. Note that if there is a non zero vector $v$ such that $Uv=0$ then $LUv=0$ which makes $LU$ singular.

Answer 2

$A$ is non-singular $\iff A^{-1}= (LU)^{-1} \exists$, but $(LU)^{-1}=U^{-1}L^{-1}$

So, $A$ is non-singular $\iff$ $U^{-1}$ and $L^{-1}$ exist $\iff$ $U$ and $L$ are invertible.