matrices

the matrices in this question have elements from the set {0,1,2,...,n}

(i) what's the probability a 2x2 matrix has determinant 0?

(ii) now consider an mxm matrix, which precisely p zero elements. what's the probability its determinant is zero?

What are the matrices?

for the 2x2 situation i think i have it mostly figured out

suppose that the each number has an equal chance of being chosen from each matrix. this means that each unique matrix has the same probability of being chosen. suppose that the elements are chosen for the first column. call the higher number a and the lower number b. then take the row with the higher number in that column and randomly choose a number for the second column in the same row, call it q. the probability that qb/a is a whole number is the probability that the higher number has all factors of q and b. now this is a difficult question for someone who has never had any number theory. that's all i can do considering i have no sense of number theory or combinatorics.

This actually seems like an interesting problem... It definitely approaches 0 as n goes to infinity, but I am at a loss as to how to determine the probability of two random pairs multiplying to the same thing.. I guess prime factorization would have to be the primary tool but finding a general formula would then seem impossible for me to do.

The mxm case would seem much easier assuming it has elements from the reals.

So... I wrote this:
for n in range(21):
        success=0
        for tryit in range(100000):
            a=random.randint(0,n)
            b=random.randint(0,n)
            c=random.randint(0,n)
            d=random.randint(0,n)
            if a*b-c*d==0:
                success+=1
        n
        1.0*success/100000



0
1.0
1
0.62429000000000001
2
0.38451000000000002
3
0.25063000000000002
4
0.18229999999999999
5
0.13222999999999999
6
0.10585
7
0.082790000000000002
8
0.068510000000000001
9
0.056000000000000001
10
0.049189999999999998
11
0.04122
12
0.037539999999999997
13
0.029929999999999998
14
0.028240000000000001
15
0.025590000000000002
16
0.022880000000000001
17
0.020320000000000001
18
0.018880000000000001
19
0.016400000000000001
20
0.016109999999999999

Sequence of the number of 2x2 matrices, elements 0..n, with determinant 0.

http://www.research.att.com/~njas/sequences/A059306

What about mmx2

>>7
1, 10, 31, 64, 113, 170, 255, 336, 449, 570, 719, 848, 1057, 1210, 1423, 1664, 1921, 2122, 2447, 2672, 3041, 3386, 3727, 4000, 4497, 4858, 5263, 5696, 6225, 6570, 7231, 7600, 8177, 8730, 9263, 9872, 10689, 11130, 11727, 12384, 13265, 13754, 14703

Divide the numbers by n^4 and you get the probability.


p n = foldl (\w a ->
         w + foldl (\x b ->
         x + foldl (\y c ->
         y + foldl (\z d ->
         z + (if a*b==c*d then 1 else 0))
       0 [0..n])
       0 [0..n])
       0 [0..n])
       0 [0..n]
For what it's worth, I used this to get the numbers and searched the database for it. So smart amirite?