Is $A$ ill conditioned matrix?

Question

Suppose we have a matrix $A$ with is its $LU$-decomposition such that $A=LU$ and suppose that $U$ is ill conditioned ($\left \| U \right \|\left \| U^{-1} \right \|$ is large) , does it mean that $A$ is ill conditioned ?

Answer 1

What if simply we consider $$A=\left[\matrix{1&0\cr0&1}\right],\quad L=\left[\matrix{10^n&0\cr0&10^{-n}}\right],\quad U=\left[\matrix{10^{-n}&0\cr0&10^{n}}\right]?$$ Clearly $L$ is lower triangular, $U$ is upper triangular, $A=LU$ and $$\hbox{cond}_2(A)=1,\quad\hbox{cond}_2(L)=\hbox{cond}_2(U)=10^{2n}.$$