I'm just starting to learn linear algebra and I'm finishing up the section of Gauss-Jordan elimination. Something that piqued my interest was the variety of different ways I could go about solving matrices. My questions are as follows:
- Is there a way to quantity the minimum number of steps it takes to solve a matrix? It is my understanding that it takes an average of $\frac{2n^3}{3}$ operations to solve an $n \times n$ matrix, according to Wikipedia. But because I'm not well versed, I'm not really sure if this is referring exactly to row operations or sheer arithmetic operations?
- For those cases where the number of steps to reduce a matrix to RREF takes longer or shorter than whatever the average count of steps is, have computer scientists or mathematicians discovered techniques for quantifying what causes this? For example, I solved a $2\times 2$ matrix in one of my problems in 4 steps when the average count of row operations (assuming I am right in interpreting what Wikipedia gives) is closer to 5. I understand that $\frac{2n^3}{3}$ is just an average, but what's actually causing the variance?
- If something about the nature of numbers or arithmetic or something is bringing about the variance, are there any ways to quantify which row operations exactly are the extraneous ones and which ones are the especially computationally effective ones? I'm sure this would be of great interest to computer systems engineers and programmers.
Don't be afraid to leave highly technical answers. I'm just a beginner but I like to use my questions as jumping-off points for future inquiries and considerations. Thanks.