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This is not an answer, but a response to this comment. Assume that $A$ is the matrix which columns are the rows that we want to test for linear independence. I am putting them are columns for no other reason than personal comfort. Doing the row transformations required in Gaussian elimination correspond to multiplying $A$ from the left by some elementary ...


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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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