The axes are in logarithmic scale, and so don't really have a zero point. What you are seeing is
$$ \log P = \min(\log I + \log \beta, \log \pi). $$
The striking part is that the slope of the line is always the same.
It's unclear what $n$ is in your question. If your matrix has dimensions $n \times n$ and your model of computation allows you to perform
basic arithmetic operations in constant time then, yes, computing the inverse matrix takes $O(n)$ time.