Assume A is a general $\displaystyle {n}\times{n}$ matrix with entries $\displaystyle \left | a_{ij} \right |\leq1$ for $\displaystyle 1\leq i,j \leq n$. Prove that the matrix $\displaystyle U$ in the $\displaystyle PA=LU$ factoriation satisfies $\displaystyle \left | a_{ij} \right |\leq 2^{n-1}$ for all $\displaystyle 1\leq i,j \leq n$.