How many arithematic operations(flops) are to $n×(n+1)$ matrix of system?

Question

Source: Linear Algebra and Its Applications David C. Lay

A system of n equations in n unknows correspond to $n×(n+1)$ augmented matrix.

One book says the reduction(elimination) to echelon form can take $2n^3/3 + n^2/2 - 7n/6$ flops, while the other book says the number of arithematic operation(flop) needs $\frac {n^3-n}{3}$ Which is correct?

Source: Linear Algebra and Its Applications Gilbert Strang

[Added. I checked Big Oh notation as you mentioned] []3
Source: Beginning Number Theory, Neville Robbins

So since you said in the comment that the number is $O(n^3)$ the following would be true:
[The number of flops = $O(n^3)]≡∃[k( with k>0) ∧ n^3]$ such that [the number of flops] $< kn^3$ for all n.

I think that's true in the case of $\frac {n^3-n}{3}$ flops, but how that's true in case of $2n^3/3 + n^2/2 - 7n/6$ flops?

Answer 1

The books count different things. First one consider * and - separately, while the second says that multiply-subtract is a single operation. Hence the discrepancy.

NB: some processors do have MAC (multiply-and-accumulate) instruction, so the Gilbert Strang's claim is not at all unfounded.

Answer 2

The amount of flops for a process depends solely on the algorithm employed in the process. Perhaps for this reason, both algorithms are computed in different flops. Still, it's possible to set an upper and lower limit of flops for both algorithms, and get an idea of computation of the algorithm in the worst case or the best case. To this, Landau symbols are used: big-O $(\mathcal{O})$ and little-o ($\mathcal{o}$).

I recommend you read this link: Family of Bachmann–Landau notations.