Finding a permutation matrix associated to a non-singular matrix

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Let $A$ be a non-singular complex matrix. Then there exist a decomposition A=LPU where L is lower triangular, U is upper triangular and P is a uniquely determined permutation matrix just depending on A. Similarly, there exists a decomposition A=L' P' L' , where L' is a lower triangular matrix and P' is a uniquely determined permutation just depending on A. (see for example https://link.springer.com/chapter/10.... theorem 1.1.10)

Question: Is there a quick way to determine the permutation matrices P and P' for a given non-singular matrix A (we can assume that A has integer entries if that helps)?