When $PA=LU$ should be used?

Question

I understand that in the case of a matrix when elimination includes a division by $\varepsilon$ one would use partial pivoting to overcome this problem.

Given a matrix, how one can tell whereas $PA=LU$ should be used or just $LU$?

Answer 1

Hint: Ask yourself whether you can row reduce \begin{align} \begin{bmatrix} 0 & 2\\ 9 & 1 \end{bmatrix} \end{align} to reduced echelon form without row exchanging.