invertible 4 × 4 matrices
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From: Timothy Murphy on 9 May 2010 09:34 quadratic wrote: > OK so far, now about the "and so on"--how many possibilities are there > for the third row (any vector not a linear combination of the first > two)? And the same for the last row? Thanks again. > >> > How does one determine how many invertible 4 × 4 matrices are there >> > with entries in Z(5) (the integers mod 5)? Thanks. >> >> Probably a bit of geometry over F_5 = Z/(5). >> First row can be any non zero vector: no = 5^4 - 1. >> Second row can be any vector not a scalar multiple of the first: >> no = 5^4 - 5. >> And so on. The next row must not be a linear combination of the first two, ie it must not lie in a 2-dimensional vector subspace. The number of elements in an r-dimensional vector space over F_q is q^r. |