Demonstrate that a matrix has no LU factorization Have to show that 
$$\begin{bmatrix}0 & 1\\1 & 1\end{bmatrix}$$
has no LU factorization.
It seems trivial just to say that this cannot have an LU decomposition because it is a lower triangular matrix already.
What is a more formal way of suggesting that this cannot be furth decomposed into LU?

Your matrix is not lower triangular. If it were, then taking $U$ to be the identity matrix would give you an LU decomposition. 
What happens here is that your matrix is non-singular, and its $1,1$ entry is zero. If you write it as $LU$ with $L$ lower triangular and $U$ upper triangular, then $A_{11}=L_{11}U_{11}$. So either $L_{11}=0$ or $U_{11}=0$, and this would made the corresponding matrix singular, and so $A$ would be singular, a contradiction. 
