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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.


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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.


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