Condition number and $LU$ decomposition

Hint:

Let’s drop the permutation matrix (assume $A=LU$) and note that the partial pivoting implies that the absolute values of the elements under the unit diagonal of $L$ are bounded from above by $1$.

Now try using this: $$ |u_{ii}^{-1}|=|e_i^TU^{-1}e_i| =|e_i^TA^{-1}Le_i| \leq\|A^{-T}e_i\|_1\|Le_i\|_\infty \leq\|A^{-T}\|_1 =\|A^{-1}\|_\infty. $$

We used the facts that $|x^Ty|\leq\|x\|_1\|y\|_\infty$ for vectors and $\|X^T\|_1=\|X\|_\infty$ for matrices.