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Wolfgang
Wolfgang
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If you do not like your own example, then you may not like this one either, but some of your students might find it interesting. I discovered it along with Roger House when he was an undergraduate.

Let F1$F_1$ be the 1x1 matrix 1$1$, and create by augmentation 0-1 matrices of larger dimension as follows: (I love < PRE > tags!)

         1 1 0 0 ... 0
         0
         1                                1 1 0 0
F_(n+1)= .0    F_n                         0                        1 1 0 0
         0                                0 1 1 0
         .:                   1 1          1 0 1 1
         .0             , so  0 1 is F_2,  0 1 0 1 is F_4, and so on.

Then det(F_n) = fib(n)$\det(F_n) = fib(n)$, which is easy. What is a little harder is that you can toggle the bits of F_n$F_n$ to get a 0-1 matrix with determinant k$k$, for any prescribed k$k$ with 0 <= k <= fib(n)$0 \le k \le fib(n)$.

Miodrag Zivkovic liked a similar example enough to include it in his paper at http://arXiv.org/abs/math.CO/0511636 . You might check out his paper to see if that example is the sort of thing for your students.

Gerhard "Ask Me About System Design" Paseman, 2010.01.15

If you do not like your own example, then you may not like this one either, but some of your students might find it interesting. I discovered it along with Roger House when he was an undergraduate.

Let F1 be the 1x1 matrix 1, and create by augmentation 0-1 matrices of larger dimension as follows: (I love < PRE > tags!)

         1 1 0 0 ... 0
         0
         1                                1 1 0 0
F_(n+1)= .    F_n                         0 1 1 0
         .                   1 1          1 0 1 1
         .             , so  0 1 is F_2,  0 1 0 1 is F_4, and so on.

Then det(F_n) = fib(n), which is easy. What is a little harder is that you can toggle the bits of F_n to get a 0-1 matrix with determinant k, for any prescribed k with 0 <= k <= fib(n).

Miodrag Zivkovic liked a similar example enough to include it in his paper at http://arXiv.org/abs/math.CO/0511636 . You might check out his paper to see if that example is the sort of thing for your students.

Gerhard "Ask Me About System Design" Paseman, 2010.01.15

If you do not like your own example, then you may not like this one either, but some of your students might find it interesting. I discovered it along with Roger House when he was an undergraduate.

Let $F_1$ be the 1x1 matrix $1$, and create by augmentation 0-1 matrices of larger dimension as follows: (I love < PRE > tags!)

         1 1 0 0 ... 0
         0
         1
F_(n+1)= 0    F_n                         1 1 0 0
         0                                0 1 1 0
         :                   1 1          1 0 1 1
         0             , so  0 1 is F_2,  0 1 0 1 is F_4, and so on.

Then $\det(F_n) = fib(n)$, which is easy. What is a little harder is that you can toggle the bits of $F_n$ to get a 0-1 matrix with determinant $k$, for any prescribed $k$ with $0 \le k \le fib(n)$.

Miodrag Zivkovic liked a similar example enough to include it in his paper at http://arXiv.org/abs/math.CO/0511636 . You might check out his paper to see if that example is the sort of thing for your students.

Gerhard "Ask Me About System Design" Paseman, 2010.01.15

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Gerhard Paseman
Gerhard Paseman
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If you do not like your own example, then you may not like this one either, but some of your students might find it interesting. I discovered it along with Roger House when he was an undergraduate.

Let F1 be the 1x1 matrix 1, and create by augmentation 0-1 matrices of larger dimension as follows: (I love < PRE > tags!)

         1 1 0 0 ... 0
         0
         1                                1 1 0 0
F_(n+1)= .    F_n                         0 1 1 0
         .                   1 1          1 0 1 1
         .             , so  0 1 is F_2,  0 1 0 1 is F_4, and so on.

Then det(F_n) = fib(n), which is easy. What is a little harder is that you can toggle the bits of F_n to get a 0-1 matrix with determinant k, for any prescribed k with 0 <= k <= fib(n).

Miodrag Zivkovic liked a similar example enough to include it in his paper at http://arXiv.org/abs/math.CO/0511636 . You might check out his paper to see if that example is the sort of thing for your students.

Gerhard "Ask Me About System Design" Paseman, 2010.01.15