Can it be done analytically? I have a system I need to solve, but would need to take a determinant of a 9 by 9 matrix. Is it worth the effort, or is there a limit (in rank) above which it's not possible to solve determinants anymore?
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$\begingroup$If you plan to compute the determinant as an alternating sum of products then you'll have $n!$ terms. Not nice for $n$ even moderately large. For instance, for $n=100$, it'll take longer than the universe exists.
Your best bet is to reduce the matrix to triangular form using row operations and compute the product of diagonal terms.
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