Chapter 2.4, Problem 10E

(a) Assume that A is an $n\times n$

matrix with entries $\left|a_{ij}\right|\le 1$ for $1\le i$, $j\le n$ . Prove that the matrix U in its $\text{PA}=\text{LU}$ factorization satisfies $\left|u_{ij}\right|\le 2^{n-1}$ for all $1\le i$, $j\le n$ . See Exercise 9(b) . (b) Formulate and prove an analogous fact for an arbitrary $n\times n$

matrix A.