# Computational Complexity of A=PLU versus other methods

I am currently trying to understand how to wrap my head around the following problem -

Consider solving $$AX=B$$ for $$X$$, where $$A$$ is $$n\times n$$, and $$X$$ and $$B$$ are $$n\times m$$. There are two obvious algorithms. The first algorithm factorizes $$A=PLU$$ using Gaussian elimination and then solves for each column of $$X$$ by forward and back substitution. The second algorithm computes $$A^{-1}$$ using Gaussian elimination and then multiplies $$X=A^{-1}B$$. Count the number of flops required by each algorithm, and show that the first one requires fewer flops.

My Book refers to the complexity of LU decomposition as $$\frac{2}{3}n^3+\mathcal{O}({n^2})$$. However my book never really discusses PLU. Am I to assume that PLU and LU are the same, complexity wise?

I've read up and looked at some proofs and I can see why some Gaussian elimination algorithms have $$\frac{2}{3}n^3+\mathcal{O}({n^2})$$ complexity using 3 basic summation properties here. I'm not sure if this proof is for just Gaussian elimination or LU. Moreover, I am not sure what algorithm the book is referring to in the second part of the question. I've read that finding the inverse takes $$\mathcal{O}(n^3)$$ complexity using Gaussian elimination and that multiplication takes $$\mathcal{O}(n^3)$$ as well, but I am not sure how to continue. Any direction would be helpful.

Edit: I believe I found something that discusses this but they seem to get a different coefficient for their LU decomposition complexity - here. I got 2/3 but they are getting 4/3 I believe.