invertible 4 × 4 matrices
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From: quadratic on 6 May 2010 14:13 How does one determine how many invertible 4 × 4 matrices are there with entries in Z(5) (the integers mod 5)? Thanks.
From: Timothy Murphy on 6 May 2010 14:37 quadratic wrote: > 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. -- Timothy Murphy e-mail: gayleard /at/ eircom.net tel: +353-86-2336090, +353-1-2842366 s-mail: School of Mathematics, Trinity College, Dublin 2, Ireland
From: Robert Israel on 6 May 2010 14:39 quadratic <quadratic(a)juno.com> writes: > How does one determine how many invertible 4 =D7 4 matrices are there > with entries in Z(5) (the integers mod 5)? Thanks. Hint: How many possibilities are there for the first row? Given the first row, how many possibilities for the second? .... -- Robert Israel israel(a)math.MyUniversitysInitials.ca Department of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada
From: quadratic on 8 May 2010 16:16 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. q > quadratic wrote: > > 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. > > -- > Timothy Murphy > e-mail: gayleard /at/ eircom.net > tel: +353-86-2336090, +353-1-2842366 > s-mail: School of Mathematics, Trinity College, Dublin 2, Ireland
From: Gerry on 8 May 2010 19:27
On May 9, 6:16 am, quadratic <quadra...(a)juno.com> 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. 1. If you can't figure that out for yourself, you are not ready for this kind of material. Honestly. Think about what you've been told, try to figure out why it's true, and try to work out for yourself where to go from there. 2. Don't top-post. > > quadratic wrote: > > > 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. |