How can I prove this proposition of linear algebra?

Question

Good afternoon! I have to show this proposition:

1) Let $A \in \mathbb R^{n\times n}$ a non singular matrix and $PA=LU$, $P$ permutation matrix, $L$ a lower triangular matrix with $1$ on its diagonal and elements below the diagonal equal or lower than one, $U$ an upper triangular matrix. Show that: $$\|A\|_F\leqslant\|U\|_F\sqrt{\frac{n(n+1)}{2}}.$$

And I have to answer this question:

2) Is $$\|A\|_{\infty}=\max_{(i,j)\in\{1,\ldots,n\}^2}|A_{i,j}|$$ a matrix norm?

I think that number two is false, but I don't know how to prove that and in number one I don't have any idea... Will someone help me? Please! I´m desperate

Answer 1

For Problem 1, you probably need $P$ to be the permutation matrix corresponding to using row-pivoted LU factorization. Since $$ \|A\|_F = \|PA\|_F $$ then you can use $$ \|A\|_F \le \|L\|_F \|U\|_F. $$ Now use the fact that $L$ was determined using row-pivoted LU to give an upper bound on the entries of $L$ and therefore an upper bound on $\|L\|_F$.

For Problem 2, first perhaps check all the properties of a norm. Depending on your definition of "matrix norm", you might want to check sub-multiplicativity.

Edit: Misread problem 2.