Gaussian Elimination Mod p
For p = 2 and n = 10, find the number of invertible 10 x 10 binary matrices, modulo 10^9 + 7.
Problem Statement
This archive keeps the full statement, math, and original media on the page.
Frogs can be placed on the real number line at integer locations. Given coprime positive integers
Two frogs placed at
A non-attacking configuration is a placement of any number of frogs such that:
- one frog is placed at
; - all other frogs are placed at distinct positive integers;
- no two frogs are attacking.
Define
You are also given
Find
Problem 988: Gaussian Elimination Mod p
Mathematical Analysis
General Linear Group over Finite Fields
Theorem. The number of invertible matrices over is:
Proof. The first row can be any nonzero vector: choices. The second row must avoid the 1-dimensional span of the first: choices. The -th row must avoid the -dimensional subspace: choices.
Evaluation for
Computing step by step: … continuing gives .
Modular Reduction
.
Connection to Random Matrix Theory
Corollary. The probability that a random binary matrix is invertible is:
For : .
As : .
Derivation
Compute the product modulo .
Complexity Analysis
multiplications.
Answer
Code
Each problem page includes the exact C++ and Python source files from the local archive.
#include <bits/stdc++.h>
using namespace std;
const long long MOD = 1e9+7;
long long power(long long b, long long e, long long m) {
long long r=1; b%=m;
while(e>0){if(e&1)r=r*b%m;b=b*b%m;e>>=1;}
return r;
}
int main() {
int n = 10; long long p = 2;
long long result = 1;
for (int k = 0; k < n; k++) {
long long factor = (power(p, n, MOD) - power(p, k, MOD) + MOD) % MOD;
result = result * factor % MOD;
}
cout << result << endl;
return 0;
}
"""
Problem 988: |GL(10, 2)| mod 10^9 + 7
Compute the order of the general linear group GL(10, F_2) modulo 10^9 + 7.
GL(n, q) consists of all invertible n x n matrices over the finite field F_q.
Its order is the product: prod_{k=0}^{n-1} (q^n - q^k).
Key results:
- |GL(n, q)| = prod_{k=0}^{n-1} (q^n - q^k)
- |GL(10, 2)| = prod_{k=0}^{9} (1024 - 2^k)
- The probability of a random binary matrix being invertible
converges to prod_{k=1}^{inf} (1 - 2^{-k}) ~ 0.2888
- Answer is the exact value modulo 10^9 + 7
Methods:
1. gl_order_mod — compute |GL(n, q)| mod m using modular arithmetic
2. gl_order_exact — compute exact |GL(n, q)| for small n
3. invertibility_prob — probability a random matrix over F_q is invertible
4. factor_analysis — examine individual (q^n - q^k) factor contributions
"""
MOD = 10**9 + 7
def gl_order_mod(n, q, mod):
"""Compute |GL(n, q)| mod m = prod_{k=0}^{n-1} (q^n - q^k) mod m."""
result = 1
for k in range(n):
factor = (pow(q, n, mod) - pow(q, k, mod)) % mod
result = (result * factor) % mod
return result
def gl_order_exact(n, q):
"""Compute exact |GL(n, q)| without modular reduction."""
result = 1
qn = q ** n
for k in range(n):
result *= (qn - q ** k)
return result
def invertibility_prob(n, q):
"""P(random n x n matrix over F_q is invertible) = |GL(n,q)| / q^(n^2)."""
return gl_order_exact(n, q) / q ** (n * n)
def invertibility_limit(q, terms=200):
"""Infinite product limit: prod_{k=1}^{inf} (1 - q^{-k})."""
result = 1.0
for k in range(1, terms + 1):
result *= (1 - q ** (-k))
return result
def factor_contributions(n, q):
"""Return list of factors (q^n - q^k) for k = 0, ..., n-1."""
qn = q ** n
return [qn - q ** k for k in range(n)]
# Verification with known values
# |GL(1, 2)| = 1 (only the 1x1 matrix [1])
assert gl_order_exact(1, 2) == 1
# |GL(2, 2)| = (4-1)(4-2) = 6
assert gl_order_exact(2, 2) == 6
# |GL(3, 2)| = (8-1)(8-2)(8-4) = 7*6*4 = 168
assert gl_order_exact(3, 2) == 168
# Verify modular computation matches exact for small cases
for n in range(1, 8):
exact = gl_order_exact(n, 2)
modular = gl_order_mod(n, 2, MOD)
assert exact % MOD == modular, f"Mismatch at n={n}"
# Cross-check: |GL(10, 2)| exact mod MOD
exact_10 = gl_order_exact(10, 2)
assert exact_10 % MOD == gl_order_mod(10, 2, MOD)
# Compute answer
answer = gl_order_mod(10, 2, MOD)
print(answer)