test/determinant_of_matrix.test.cpp

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• Last update: 2025-03-10 04:06:44+08:00
• Problem: https://judge.yosupo.jp/problem/matrix_det

Depends on

• default/t.cpp
• linalg/matrixMint.cpp
• modint/MontgomeryModInt.cpp

Code

#define PROBLEM "https://judge.yosupo.jp/problem/matrix_det"

#include "../default/t.cpp"
#include "../modint/MontgomeryModInt.cpp"
#include "../linalg/matrixMint.cpp"

signed main() {
  ios::sync_with_stdio(false), cin.tie(NULL);

  int n; cin >> n;
  matrix<mint> M(n, n);
  cin >> M;
  cout << M.det() << '\n';

  return 0;
}

#line 1 "test/determinant_of_matrix.test.cpp"
#define PROBLEM "https://judge.yosupo.jp/problem/matrix_det"

#line 1 "default/t.cpp"
#include <algorithm>
#include <array>
#include <bitset>
#include <cassert>
#include <cctype>
#include <cfenv>
#include <cfloat>
#include <chrono>
#include <cinttypes>
#include <climits>
#include <cmath>
#include <complex>
#include <cstdarg>
#include <cstddef>
#include <cstdint>
#include <cstdio>
#include <cstdlib>
#include <cstring>
#include <deque>
#include <fstream>
#include <functional>
#include <initializer_list>
#include <iomanip>
#include <ios>
#include <iostream>
#include <istream>
#include <iterator>
#include <limits>
#include <list>
#include <map>
#include <memory>
#include <new>
#include <numeric>
#include <ostream>
#include <queue>
#include <random>
#include <set>
#include <sstream>
#include <stack>
#include <streambuf>
#include <string>
#include <tuple>
#include <type_traits>
#include <variant>
#include <bit>
#include <compare>
#include <concepts>
#include <numbers>
#include <ranges>
#include <span>

#define INT128_MAX (__int128)(((unsigned __int128) 1 << ((sizeof(__int128) * __CHAR_BIT__) - 1)) - 1)
#define INT128_MIN (-INT128_MAX - 1)

#define clock chrono::steady_clock::now().time_since_epoch().count()

using namespace std;

template<class T1, class T2>
ostream& operator<<(ostream& os, const pair<T1, T2> pr) {
  return os << pr.first << ' ' << pr.second;
}
template<class T, size_t N>
ostream& operator<<(ostream& os, const array<T, N> &arr) {
  for(size_t i = 0; T x : arr) {
    os << x;
    if (++i != N) os << ' ';
  }
  return os;
}
template<class T>
ostream& operator<<(ostream& os, const vector<T> &vec) {
  for(size_t i = 0; T x : vec) {
    os << x;
    if (++i != size(vec)) os << ' ';
  }
  return os;
}
template<class T>
ostream& operator<<(ostream& os, const set<T> &s) {
  for(size_t i = 0; T x : s) {
    os << x;
    if (++i != size(s)) os << ' ';
  }
  return os;
}
template<class T1, class T2>
ostream& operator<<(ostream& os, const map<T1, T2> &m) {
  for(size_t i = 0; pair<T1, T2> x : m) {
    os << x;
    if (++i != size(m)) os << ' ';
  }
  return os;
}

#ifdef DEBUG
#define dbg(...) cerr << '(', _do(#__VA_ARGS__), cerr << ") = ", _do2(__VA_ARGS__)
template<typename T> void _do(T &&x) { cerr << x; }
template<typename T, typename ...S> void _do(T &&x, S&&...y) { cerr << x << ", "; _do(y...); }
template<typename T> void _do2(T &&x) { cerr << x << endl; }
template<typename T, typename ...S> void _do2(T &&x, S&&...y) { cerr << x << ", "; _do2(y...); }
#else
#define dbg(...)
#endif

using ll = long long;
using ull = unsigned long long;
using ldb = long double;
using pii = pair<int, int>;
using pll = pair<ll, ll>;

template<typename T> using min_heap = priority_queue<T, vector<T>, greater<T>>;
template<typename T> using max_heap = priority_queue<T>;

template<ranges::forward_range rng, class T = ranges::range_value_t<rng>, class OP = plus<T>>
void pSum(rng &v) {
  if (!v.empty())
    for(T p = v[0]; T &x : v | views::drop(1))
      x = p = OP()(p, x);
}
template<ranges::forward_range rng, class T = ranges::range_value_t<rng>, class OP>
void pSum(rng &v, OP op) {
  if (!v.empty())
    for(T p = v[0]; T &x : v | views::drop(1))
      x = p = op(p, x);
}

template<ranges::forward_range rng>
void Unique(rng &v) {
  ranges::sort(v);
  v.resize(unique(v.begin(), v.end()) - v.begin());
}

template<ranges::random_access_range rng>
rng invPerm(rng p) {
  rng ret = p;
  for(int i = 0; i < ssize(p); i++)
    ret[p[i]] = i;
  return ret;
}

template<ranges::random_access_range rng, ranges::random_access_range rng2>
rng Permute(rng v, rng2 p) {
  rng ret = v;
  for(int i = 0; i < ssize(p); i++)
    ret[p[i]] = v[i];
  return ret;
}

template<bool directed>
vector<vector<int>> readGraph(int n, int m, int base) {
  vector<vector<int>> g(n);
  for(int i = 0; i < m; i++) {
    int u, v; cin >> u >> v;
    u -= base, v -= base;
    g[u].emplace_back(v);
    if constexpr (!directed)
      g[v].emplace_back(u);
  }
  return g;
}

template<class T>
void setBit(T &msk, int bit, bool x) {
  msk = (msk & ~(T(1) << bit)) | (T(x) << bit);
}
template<class T> void flipBit(T &msk, int bit) { msk ^= T(1) << bit; }
template<class T> bool getBit(T msk, int bit) { return msk >> bit & T(1); }

template<class T>
T floorDiv(T a, T b) {
  if (b < 0) a *= -1, b *= -1;
  return a >= 0 ? a / b : (a - b + 1) / b;
}
template<class T>
T ceilDiv(T a, T b) {
  if (b < 0) a *= -1, b *= -1;
  return a >= 0 ? (a + b - 1) / b : a / b;
}

template<class T> bool chmin(T &a, T b) { return a > b ? a = b, 1 : 0; }
template<class T> bool chmax(T &a, T b) { return a < b ? a = b, 1 : 0; }
#line 1 "modint/MontgomeryModInt.cpp"
//reference: https://github.com/NyaanNyaan/library/blob/master/modint/montgomery-modint.hpp#L10
//note: mod should be an odd prime less than 2^30.

template<uint32_t mod>
struct MontgomeryModInt {
  using mint = MontgomeryModInt;
  using i32 = int32_t;
  using u32 = uint32_t;
  using u64 = uint64_t;

  static constexpr u32 get_r() {
    u32 res = 1, base = mod;
    for(i32 i = 0; i < 31; i++)
      res *= base, base *= base;
    return -res;
  }

  static constexpr u32 get_mod() {
    return mod;
  }

  static constexpr u32 n2 = -u64(mod) % mod; //2^64 % mod
  static constexpr u32 r = get_r(); //-P^{-1} % 2^32

  u32 a;

  static u32 reduce(const u64 &b) {
    return (b + u64(u32(b) * r) * mod) >> 32;
  }

  static u32 transform(const u64 &b) {
    return reduce(u64(b) * n2);
  }

  MontgomeryModInt() : a(0) {}
  MontgomeryModInt(const int64_t &b) 
    : a(transform(b % mod + mod)) {}

  mint pow(u64 k) const {
    mint res(1), base(*this);
    while(k) {
      if (k & 1) 
        res *= base;
      base *= base, k >>= 1;
    }
    return res;
  }

  mint inverse() const { return (*this).pow(mod - 2); }

  u32 get() const {
    u32 res = reduce(a);
    return res >= mod ? res - mod : res;
  }

  mint& operator+=(const mint &b) {
    if (i32(a += b.a - 2 * mod) < 0) a += 2 * mod;
    return *this;
  }

  mint& operator-=(const mint &b) {
    if (i32(a -= b.a) < 0) a += 2 * mod;
    return *this;
  }

  mint& operator*=(const mint &b) {
    a = reduce(u64(a) * b.a);
    return *this;
  }

  mint& operator/=(const mint &b) {
    a = reduce(u64(a) * b.inverse().a);
    return *this;
  }

  mint operator-() { return mint() - mint(*this); }
  bool operator==(mint b) const {
    return (a >= mod ? a - mod : a) == (b.a >= mod ? b.a - mod : b.a);
  }
  bool operator!=(mint b) const {
    return (a >= mod ? a - mod : a) != (b.a >= mod ? b.a - mod : b.a);
  }

  friend mint operator+(mint c, mint d) { return c += d; }
  friend mint operator-(mint c, mint d) { return c -= d; }
  friend mint operator*(mint c, mint d) { return c *= d; }
  friend mint operator/(mint c, mint d) { return c /= d; }

  friend ostream& operator<<(ostream& os, const mint& b) {
    return os << b.get();
  }
  friend istream& operator>>(istream& is, mint& b) {
    int64_t val;
    is >> val;
    b = mint(val);
    return is;
  }
};

using mint = MontgomeryModInt<998244353>;
#line 1 "linalg/matrixMint.cpp"
template<class Mint>
struct matrix : vector<vector<Mint>> {
  matrix(int n, int m) : vector<vector<Mint>>(n, vector<Mint>(m, 0)) {}
  matrix(int n) : vector<vector<Mint>>(n, vector<Mint>(n, 0)) {}

  int n() const { return ssize(*this); }
  int m() const { return n() == 0 ? 0 : ssize((*this)[0]); }

  static matrix I(int n) {
    auto res = matrix(n, n);
    for(int i = 0; i < n; i++)
      res[i][i] = 1;
    return res;
  }

  matrix& operator+=(const matrix &b) {
    assert(n() == b.n());
    assert(m() == b.m());
    for(int i = 0; i < n(); i++)
      for(int j = 0; j < m(); j++)
        (*this)[i][j] += b[i][j];
    return *this;
  }

  matrix& operator-=(const matrix &b) {
    assert(n() == b.n());
    assert(m() == b.m());
    for(int i = 0; i < n(); i++)
      for(int j = 0; j < m(); j++)
        (*this)[i][j] -= b[i][j];
    return *this;
  }

  matrix& operator*=(const matrix &b) {
    assert(m() == b.n());
    auto res = matrix(n(), b.m());
    for(int i = 0; i < n(); i++)
      for(int k = 0; k < m(); k++)
        for(int j = 0; j < b.m(); j++)
          res[i][j] += (*this)[i][k] * b[k][j];
    this -> swap(res);
    return *this;
  }

  matrix pow(ll k) const {
    assert(n() == m());
    auto res = I(n()), base = *this;
    while(k) {
      if (k & 1) res *= base;
      base *= base, k >>= 1;
    }
    return res;
  }

  tuple<matrix, vector<int>, int> eliminate() {
    int sgn = 1;
    matrix M = *this;
    vector<int> pivot_row;
    for(int row = 0, col = 0; row < n() and col < m(); col++) {
      int p_row = -1;
      for(int i = row; i < n() and p_row == -1; i++)
        if (M[i][col] != 0) 
          p_row = i;
      if (p_row == -1) continue;
      pivot_row.emplace_back(row);
      if (row != p_row) {
        for(int j = col; j < m(); j++)
          swap(M[row][j], M[p_row][j]);
        sgn *= -1;
      }
      for(int i = 0; i < n(); i++) {
        if (i == row or M[i][col] == 0) continue;
        Mint s = M[i][col] / M[row][col];
        for(int j = col; j < m(); j++)
          M[i][j] -= M[row][j] * s;
      }
      row++;
    }
    return {M, pivot_row, sgn};
  }

  Mint det() {
    assert(n() == m());
    auto [M, pr, sgn] = eliminate();
    if (ssize(pr) != n()) {
      return Mint(0);
    } else {
      Mint d = sgn;
      for(int i = 0; i < n(); i++)
        d *= M[i][i];
      return d;
    }
  }

  int rank() {
    return get<1>(eliminate()).size();
  }

  pair<bool, matrix> inv() {
    assert(n() == m());
    matrix M(n(), 2 * n());
    for(int i = 0; i < n(); i++) {
      for(int j = 0; j < n(); j++)
        M[i][j] = (*this)[i][j];
      M[i][n() + i] = 1;
    }
    matrix tmp = get<0>(M.eliminate());
    matrix MI(n(), n());
    for(int i = 0; i < n(); i++) {
      if (tmp[i][i] == 0) return {false, matrix(0, 0)};
      Mint r = tmp[i][i].inverse();
      for(int j = 0; j < n(); j++)
        MI[i][j] = tmp[i][j + n()] * r;
    }
    return {true, MI};
  }

  pair<vector<Mint>, matrix> solve_linear(vector<Mint> b) {
    assert(n() == ssize(b));

    matrix M(n(), m() + 1);
    for(int i = 0; i < n(); i++) {
      for(int j = 0; j < m(); j++)
        M[i][j] = (*this)[i][j];
      M[i][m()] = b[i];
    }

    auto [N, pr, _] = M.eliminate();
    vector<Mint> x(m());
    vector<int> where(m(), -1), inv_where(m(), -1);
    for(int row : pr) {
      int col = 0;
      while(N[row][col] == 0) col++;
      if (col < m())
        where[col] = row, inv_where[row] = col;
    }

    for(int i = 0; i < m(); i++)
      if (where[i] != -1)
        x[i] = N[where[i]][m()] / N[where[i]][i];

    for(int i = 0; i < n(); i++) {
      Mint s = -N[i][m()];
      for(int j = 0; j < m(); j++)
        s += x[j] * N[i][j];
      if (s != Mint(0))
        return {vector<Mint>(), matrix(0)};
    }

    matrix basis(m() - ssize(pr), m());
    for(int col = 0, last_row = 0, k = 0; col < m(); col++) {
      if (where[col] != -1) {
        last_row = where[col];
      } else {
        basis[k][col] = 1;
        for(int i = 0; i <= last_row; i++)
          basis[k][inv_where[i]] = -N[i][col] / N[i][inv_where[i]];
        k++;
      }
    }

    return {x, basis};
  }

  matrix operator-() { return matrix(n(), m()) - (*this); }
  
  friend matrix operator+(matrix a, matrix b) { return a += b; }
  friend matrix operator-(matrix a, matrix b) { return a -= b; }
  friend matrix operator*(matrix a, matrix b) { return a *= b; }
  
  friend ostream& operator<<(ostream& os, const matrix& b) {
    for(int i = 0; i < b.n(); i++) {
      os << '\n';
      for(int j = 0; j < b.m(); j++)
        os << b[i][j] << ' ';
    }
    return os;
  }
  friend istream& operator>>(istream& is, matrix& b) {
    for(int i = 0; i < b.n(); i++)
      for(int j = 0; j < b.m(); j++)
        is >> b[i][j];
    return is;
  }
};
#line 6 "test/determinant_of_matrix.test.cpp"

signed main() {
  ios::sync_with_stdio(false), cin.tie(NULL);

  int n; cin >> n;
  matrix<mint> M(n, n);
  cin >> M;
  cout << M.det() << '\n';

  return 0;
}

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