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#define PROBLEM "https://judge.yosupo.jp/problem/matrix_det" #include <bits/stdc++.h> #include "../../atcoder/modint.hpp" using namespace std; using namespace atcoder; #include "../Matrix.h" #include "../../buffered_reader.h" #define REP(i, a) for (int i = 0, _##i = (a); i < _##i; ++i) int32_t main() { ios::sync_with_stdio(0); cin.tie(0); int n = IO::get<int>(); Matrix<modint998244353> a(n, n); REP(i,n) REP(j,n) { int x = IO::get<int>(); a[i][j] = x; } auto tmp = a.gauss(); cout << tmp.det().val() << endl; return 0; }
#line 1 "Math/tests/matrix_det.test.cpp" #define PROBLEM "https://judge.yosupo.jp/problem/matrix_det" #include <bits/stdc++.h> #line 1 "atcoder/modint.hpp" #line 6 "atcoder/modint.hpp" #include <type_traits> #ifdef _MSC_VER #include <intrin.h> #endif #line 1 "atcoder/internal_math.hpp" #line 5 "atcoder/internal_math.hpp" #ifdef _MSC_VER #include <intrin.h> #endif namespace atcoder { namespace internal { // @param m `1 <= m` // @return x mod m constexpr long long safe_mod(long long x, long long m) { x %= m; if (x < 0) x += m; return x; } // Fast modular multiplication by barrett reduction // Reference: https://en.wikipedia.org/wiki/Barrett_reduction // NOTE: reconsider after Ice Lake struct barrett { unsigned int _m; unsigned long long im; // @param m `1 <= m < 2^31` explicit barrett(unsigned int m) : _m(m), im((unsigned long long)(-1) / m + 1) {} // @return m unsigned int umod() const { return _m; } // @param a `0 <= a < m` // @param b `0 <= b < m` // @return `a * b % m` unsigned int mul(unsigned int a, unsigned int b) const { // [1] m = 1 // a = b = im = 0, so okay // [2] m >= 2 // im = ceil(2^64 / m) // -> im * m = 2^64 + r (0 <= r < m) // let z = a*b = c*m + d (0 <= c, d < m) // a*b * im = (c*m + d) * im = c*(im*m) + d*im = c*2^64 + c*r + d*im // c*r + d*im < m * m + m * im < m * m + 2^64 + m <= 2^64 + m * (m + 1) < 2^64 * 2 // ((ab * im) >> 64) == c or c + 1 unsigned long long z = a; z *= b; #ifdef _MSC_VER unsigned long long x; _umul128(z, im, &x); #else unsigned long long x = (unsigned long long)(((unsigned __int128)(z)*im) >> 64); #endif unsigned int v = (unsigned int)(z - x * _m); if (_m <= v) v += _m; return v; } }; // @param n `0 <= n` // @param m `1 <= m` // @return `(x ** n) % m` constexpr long long pow_mod_constexpr(long long x, long long n, int m) { if (m == 1) return 0; unsigned int _m = (unsigned int)(m); unsigned long long r = 1; unsigned long long y = safe_mod(x, m); while (n) { if (n & 1) r = (r * y) % _m; y = (y * y) % _m; n >>= 1; } return r; } // Reference: // M. Forisek and J. Jancina, // Fast Primality Testing for Integers That Fit into a Machine Word // @param n `0 <= n` constexpr bool is_prime_constexpr(int n) { if (n <= 1) return false; if (n == 2 || n == 7 || n == 61) return true; if (n % 2 == 0) return false; long long d = n - 1; while (d % 2 == 0) d /= 2; constexpr long long bases[3] = {2, 7, 61}; for (long long a : bases) { long long t = d; long long y = pow_mod_constexpr(a, t, n); while (t != n - 1 && y != 1 && y != n - 1) { y = y * y % n; t <<= 1; } if (y != n - 1 && t % 2 == 0) { return false; } } return true; } template <int n> constexpr bool is_prime = is_prime_constexpr(n); // @param b `1 <= b` // @return pair(g, x) s.t. g = gcd(a, b), xa = g (mod b), 0 <= x < b/g constexpr std::pair<long long, long long> inv_gcd(long long a, long long b) { a = safe_mod(a, b); if (a == 0) return {b, 0}; // Contracts: // [1] s - m0 * a = 0 (mod b) // [2] t - m1 * a = 0 (mod b) // [3] s * |m1| + t * |m0| <= b long long s = b, t = a; long long m0 = 0, m1 = 1; while (t) { long long u = s / t; s -= t * u; m0 -= m1 * u; // |m1 * u| <= |m1| * s <= b // [3]: // (s - t * u) * |m1| + t * |m0 - m1 * u| // <= s * |m1| - t * u * |m1| + t * (|m0| + |m1| * u) // = s * |m1| + t * |m0| <= b auto tmp = s; s = t; t = tmp; tmp = m0; m0 = m1; m1 = tmp; } // by [3]: |m0| <= b/g // by g != b: |m0| < b/g if (m0 < 0) m0 += b / s; return {s, m0}; } // Compile time primitive root // @param m must be prime // @return primitive root (and minimum in now) constexpr int primitive_root_constexpr(int m) { if (m == 2) return 1; if (m == 167772161) return 3; if (m == 469762049) return 3; if (m == 754974721) return 11; if (m == 998244353) return 3; int divs[20] = {}; divs[0] = 2; int cnt = 1; int x = (m - 1) / 2; while (x % 2 == 0) x /= 2; for (int i = 3; (long long)(i)*i <= x; i += 2) { if (x % i == 0) { divs[cnt++] = i; while (x % i == 0) { x /= i; } } } if (x > 1) { divs[cnt++] = x; } for (int g = 2;; g++) { bool ok = true; for (int i = 0; i < cnt; i++) { if (pow_mod_constexpr(g, (m - 1) / divs[i], m) == 1) { ok = false; break; } } if (ok) return g; } } template <int m> constexpr int primitive_root = primitive_root_constexpr(m); // @param n `n < 2^32` // @param m `1 <= m < 2^32` // @return sum_{i=0}^{n-1} floor((ai + b) / m) (mod 2^64) unsigned long long floor_sum_unsigned(unsigned long long n, unsigned long long m, unsigned long long a, unsigned long long b) { unsigned long long ans = 0; while (true) { if (a >= m) { ans += n * (n - 1) / 2 * (a / m); a %= m; } if (b >= m) { ans += n * (b / m); b %= m; } unsigned long long y_max = a * n + b; if (y_max < m) break; // y_max < m * (n + 1) // floor(y_max / m) <= n n = (unsigned long long)(y_max / m); b = (unsigned long long)(y_max % m); std::swap(m, a); } return ans; } } // namespace internal } // namespace atcoder #line 1 "atcoder/internal_type_traits.hpp" #line 7 "atcoder/internal_type_traits.hpp" namespace atcoder { namespace internal { #ifndef _MSC_VER template <class T> using is_signed_int128 = typename std::conditional<std::is_same<T, __int128_t>::value || std::is_same<T, __int128>::value, std::true_type, std::false_type>::type; template <class T> using is_unsigned_int128 = typename std::conditional<std::is_same<T, __uint128_t>::value || std::is_same<T, unsigned __int128>::value, std::true_type, std::false_type>::type; template <class T> using make_unsigned_int128 = typename std::conditional<std::is_same<T, __int128_t>::value, __uint128_t, unsigned __int128>; template <class T> using is_integral = typename std::conditional<std::is_integral<T>::value || is_signed_int128<T>::value || is_unsigned_int128<T>::value, std::true_type, std::false_type>::type; template <class T> using is_signed_int = typename std::conditional<(is_integral<T>::value && std::is_signed<T>::value) || is_signed_int128<T>::value, std::true_type, std::false_type>::type; template <class T> using is_unsigned_int = typename std::conditional<(is_integral<T>::value && std::is_unsigned<T>::value) || is_unsigned_int128<T>::value, std::true_type, std::false_type>::type; template <class T> using to_unsigned = typename std::conditional< is_signed_int128<T>::value, make_unsigned_int128<T>, typename std::conditional<std::is_signed<T>::value, std::make_unsigned<T>, std::common_type<T>>::type>::type; #else template <class T> using is_integral = typename std::is_integral<T>; template <class T> using is_signed_int = typename std::conditional<is_integral<T>::value && std::is_signed<T>::value, std::true_type, std::false_type>::type; template <class T> using is_unsigned_int = typename std::conditional<is_integral<T>::value && std::is_unsigned<T>::value, std::true_type, std::false_type>::type; template <class T> using to_unsigned = typename std::conditional<is_signed_int<T>::value, std::make_unsigned<T>, std::common_type<T>>::type; #endif template <class T> using is_signed_int_t = std::enable_if_t<is_signed_int<T>::value>; template <class T> using is_unsigned_int_t = std::enable_if_t<is_unsigned_int<T>::value>; template <class T> using to_unsigned_t = typename to_unsigned<T>::type; } // namespace internal } // namespace atcoder #line 14 "atcoder/modint.hpp" namespace atcoder { namespace internal { struct modint_base {}; struct static_modint_base : modint_base {}; template <class T> using is_modint = std::is_base_of<modint_base, T>; template <class T> using is_modint_t = std::enable_if_t<is_modint<T>::value>; } // namespace internal template <int m, std::enable_if_t<(1 <= m)>* = nullptr> struct static_modint : internal::static_modint_base { using mint = static_modint; public: static constexpr int mod() { return m; } static mint raw(int v) { mint x; x._v = v; return x; } static_modint() : _v(0) {} template <class T, internal::is_signed_int_t<T>* = nullptr> static_modint(T v) { long long x = (long long)(v % (long long)(umod())); if (x < 0) x += umod(); _v = (unsigned int)(x); } template <class T, internal::is_unsigned_int_t<T>* = nullptr> static_modint(T v) { _v = (unsigned int)(v % umod()); } unsigned int val() const { return _v; } mint& operator++() { _v++; if (_v == umod()) _v = 0; return *this; } mint& operator--() { if (_v == 0) _v = umod(); _v--; return *this; } mint operator++(int) { mint result = *this; ++*this; return result; } mint operator--(int) { mint result = *this; --*this; return result; } mint& operator+=(const mint& rhs) { _v += rhs._v; if (_v >= umod()) _v -= umod(); return *this; } mint& operator-=(const mint& rhs) { _v -= rhs._v; if (_v >= umod()) _v += umod(); return *this; } mint& operator*=(const mint& rhs) { unsigned long long z = _v; z *= rhs._v; _v = (unsigned int)(z % umod()); return *this; } mint& operator/=(const mint& rhs) { return *this = *this * rhs.inv(); } mint operator+() const { return *this; } mint operator-() const { return mint() - *this; } mint pow(long long n) const { assert(0 <= n); mint x = *this, r = 1; while (n) { if (n & 1) r *= x; x *= x; n >>= 1; } return r; } mint inv() const { if (prime) { assert(_v); return pow(umod() - 2); } else { auto eg = internal::inv_gcd(_v, m); assert(eg.first == 1); return eg.second; } } friend mint operator+(const mint& lhs, const mint& rhs) { return mint(lhs) += rhs; } friend mint operator-(const mint& lhs, const mint& rhs) { return mint(lhs) -= rhs; } friend mint operator*(const mint& lhs, const mint& rhs) { return mint(lhs) *= rhs; } friend mint operator/(const mint& lhs, const mint& rhs) { return mint(lhs) /= rhs; } friend bool operator==(const mint& lhs, const mint& rhs) { return lhs._v == rhs._v; } friend bool operator!=(const mint& lhs, const mint& rhs) { return lhs._v != rhs._v; } private: unsigned int _v; static constexpr unsigned int umod() { return m; } static constexpr bool prime = internal::is_prime<m>; }; template <int id> struct dynamic_modint : internal::modint_base { using mint = dynamic_modint; public: static int mod() { return (int)(bt.umod()); } static void set_mod(int m) { assert(1 <= m); bt = internal::barrett(m); } static mint raw(int v) { mint x; x._v = v; return x; } dynamic_modint() : _v(0) {} template <class T, internal::is_signed_int_t<T>* = nullptr> dynamic_modint(T v) { long long x = (long long)(v % (long long)(mod())); if (x < 0) x += mod(); _v = (unsigned int)(x); } template <class T, internal::is_unsigned_int_t<T>* = nullptr> dynamic_modint(T v) { _v = (unsigned int)(v % mod()); } unsigned int val() const { return _v; } mint& operator++() { _v++; if (_v == umod()) _v = 0; return *this; } mint& operator--() { if (_v == 0) _v = umod(); _v--; return *this; } mint operator++(int) { mint result = *this; ++*this; return result; } mint operator--(int) { mint result = *this; --*this; return result; } mint& operator+=(const mint& rhs) { _v += rhs._v; if (_v >= umod()) _v -= umod(); return *this; } mint& operator-=(const mint& rhs) { _v += mod() - rhs._v; if (_v >= umod()) _v -= umod(); return *this; } mint& operator*=(const mint& rhs) { _v = bt.mul(_v, rhs._v); return *this; } mint& operator/=(const mint& rhs) { return *this = *this * rhs.inv(); } mint operator+() const { return *this; } mint operator-() const { return mint() - *this; } mint pow(long long n) const { assert(0 <= n); mint x = *this, r = 1; while (n) { if (n & 1) r *= x; x *= x; n >>= 1; } return r; } mint inv() const { auto eg = internal::inv_gcd(_v, mod()); assert(eg.first == 1); return eg.second; } friend mint operator+(const mint& lhs, const mint& rhs) { return mint(lhs) += rhs; } friend mint operator-(const mint& lhs, const mint& rhs) { return mint(lhs) -= rhs; } friend mint operator*(const mint& lhs, const mint& rhs) { return mint(lhs) *= rhs; } friend mint operator/(const mint& lhs, const mint& rhs) { return mint(lhs) /= rhs; } friend bool operator==(const mint& lhs, const mint& rhs) { return lhs._v == rhs._v; } friend bool operator!=(const mint& lhs, const mint& rhs) { return lhs._v != rhs._v; } private: unsigned int _v; static internal::barrett bt; static unsigned int umod() { return bt.umod(); } }; template <int id> internal::barrett dynamic_modint<id>::bt(998244353); using modint998244353 = static_modint<998244353>; using modint1000000007 = static_modint<1000000007>; using modint = dynamic_modint<-1>; namespace internal { template <class T> using is_static_modint = std::is_base_of<internal::static_modint_base, T>; template <class T> using is_static_modint_t = std::enable_if_t<is_static_modint<T>::value>; template <class> struct is_dynamic_modint : public std::false_type {}; template <int id> struct is_dynamic_modint<dynamic_modint<id>> : public std::true_type {}; template <class T> using is_dynamic_modint_t = std::enable_if_t<is_dynamic_modint<T>::value>; } // namespace internal } // namespace atcoder #line 5 "Math/tests/matrix_det.test.cpp" using namespace std; using namespace atcoder; #line 1 "Math/Matrix.h" // Matrix, which works for both double and int {{{ // Copied partially from https://judge.yosupo.jp/submission/54653 // // Tested: // - (mat mul): https://judge.yosupo.jp/problem/matrix_product // - (mat pow): https://oj.vnoi.info/problem/icpc21_mt_k // - (mat pow): https://oj.vnoi.info/problem/icpc21_mb_h // - (gauss): https://oj.vnoi.info/problem/vmrook // - (inverse): https://oj.vnoi.info/problem/dtl_lsr // - (inverse): https://judge.yosupo.jp/problem/inverse_matrix // - (det): https://judge.yosupo.jp/problem/matrix_det template<typename T> struct Matrix { int n_row, n_col; vector<T> x; // accessors typename vector<T>::iterator operator [] (int r) { return x.begin() + r * n_col; } inline T get(int i, int j) const { return x[i * n_col + j]; } vector<T> at(int r) const { return vector<T> { x.begin() + r * n_col, x.begin() + (r+1) * n_col }; } // constructors Matrix() = default; Matrix(int _n_row, int _n_col) : n_row(_n_row), n_col(_n_col), x(n_row * n_col) {} Matrix(const vector<vector<T>>& d) : n_row(d.size()), n_col(d.size() ? d[0].size() : 0) { for (auto& row : d) std::copy(row.begin(), row.end(), std::back_inserter(x)); } // convert to 2d vec vector<vector<T>> vecvec() const { vector<vector<T>> ret(n_row); for (int i = 0; i < n_row; i++) { std::copy(x.begin() + i*n_col, x.begin() + (i+1)*n_col, std::back_inserter(ret[i])); } return ret; } operator vector<vector<T>>() const { return vecvec(); } static Matrix identity(int n) { Matrix res(n, n); for (int i = 0; i < n; i++) { res[i][i] = 1; } return res; } Matrix transpose() const { Matrix res(n_col, n_row); for (int i = 0; i < n_row; i++) { for (int j = 0; j < n_col; j++) { res[j][i] = this->get(i, j); } } return res; } Matrix& operator *= (const Matrix& r) { return *this = *this * r; } Matrix operator * (const Matrix& r) const { assert(n_col == r.n_row); Matrix res(n_row, r.n_col); for (int i = 0; i < n_row; i++) { for (int k = 0; k < n_col; k++) { for (int j = 0; j < r.n_col; j++) { res[i][j] += this->get(i, k) * r.get(k, j); } } } return res; } Matrix pow(long long n) const { assert(n_row == n_col); Matrix res = identity(n_row); if (n == 0) return res; bool res_is_id = true; for (int i = 63 - __builtin_clzll(n); i >= 0; i--) { if (!res_is_id) res *= res; if ((n >> i) & 1) res *= (*this), res_is_id = false; } return res; } // Gauss template <typename T2, typename std::enable_if<std::is_floating_point<T2>::value>::type * = nullptr> static int choose_pivot(const Matrix<T2> &mtr, int h, int c) noexcept { int piv = -1; for (int j = h; j < mtr.n_row; j++) { if (mtr.get(j, c) and (piv < 0 or std::abs(mtr.get(j, c)) > std::abs(mtr.get(piv, c)))) piv = j; } return piv; } template <typename T2, typename std::enable_if<!std::is_floating_point<T2>::value>::type * = nullptr> static int choose_pivot(const Matrix<T2> &mtr, int h, int c) noexcept { for (int j = h; j < mtr.n_row; j++) { if (mtr.get(j, c) != T(0)) return j; } return -1; } // return upper triangle matrix [[nodiscard]] Matrix gauss() const { int c = 0; Matrix mtr(*this); vector<int> ws; ws.reserve(n_col); for (int h = 0; h < n_row; h++) { if (c == n_col) break; int piv = choose_pivot(mtr, h, c); if (piv == -1) { c++; h--; continue; } if (h != piv) { for (int w = 0; w < n_col; w++) { swap(mtr[piv][w], mtr[h][w]); mtr[piv][w] *= -1; // for determinant } } ws.clear(); for (int w = c; w < n_col; w++) { if (mtr[h][w] != 0) ws.emplace_back(w); } const T hcinv = T(1) / mtr[h][c]; for (int hh = 0; hh < n_row; hh++) { if (hh != h) { const T coeff = mtr[hh][c] * hcinv; for (auto w : ws) mtr[hh][w] -= mtr[h][w] * coeff; mtr[hh][c] = 0; } } c++; } return mtr; } // For upper triangle matrix T det() const { T ret = 1; for (int i = 0; i < n_row; i++) { ret *= get(i, i); } return ret; } // return rank of inverse matrix. If rank < n -> not invertible int inverse() { assert(n_row == n_col); vector<vector<T>> ret = identity(n_row), tmp = *this; int rank = 0; for (int i = 0; i < n_row; i++) { int ti = i; while (ti < n_row && tmp[ti][i] == 0) ++ti; if (ti == n_row) continue; else ++rank; ret[i].swap(ret[ti]); tmp[i].swap(tmp[ti]); T inv = T(1) / tmp[i][i]; for (int j = 0; j < n_col; j++) ret[i][j] *= inv; for (int j = i+1; j < n_col; j++) tmp[i][j] *= inv; for (int h = 0; h < n_row; h++) { if (i == h) continue; const T c = -tmp[h][i]; for (int j = 0; j < n_col; j++) ret[h][j] += ret[i][j] * c; for (int j = i+1; j < n_col; j++) tmp[h][j] += tmp[i][j] * c; } } *this = ret; return rank; } // sum of all elements in this matrix T sum_all() { return submatrix_sum(0, 0, n_row, n_col); } // sum of [r1, r2) x [c1, c2) T submatrix_sum(int r1, int c1, int r2, int c2) { T res {0}; for (int r = r1; r < r2; ++r) { res += std::accumulate( x.begin() + r * n_col + c1, x.begin() + r * n_col + c2, T{0}); } return res; } }; template<typename T> ostream& operator << (ostream& cout, const Matrix<T>& m) { cout << m.n_row << ' ' << m.n_col << endl; for (int i = 0; i < m.n_row; ++i) { cout << "row [" << i << "] = " << m.at(i) << endl; } return cout; } // }}} #line 1 "buffered_reader.h" // Buffered reader {{{ namespace IO { const int BUFSIZE = 1<<14; char buf[BUFSIZE + 1], *inp = buf; bool reacheof; char get_char() { if (!*inp && !reacheof) { memset(buf, 0, sizeof buf); int tmp = fread(buf, 1, BUFSIZE, stdin); if (tmp != BUFSIZE) reacheof = true; inp = buf; } return *inp++; } template<typename T> T get() { int neg = 0; T res = 0; char c = get_char(); while (!std::isdigit(c) && c != '-' && c != '+') c = get_char(); if (c == '+') { neg = 0; } else if (c == '-') { neg = 1; } else res = c - '0'; c = get_char(); while (std::isdigit(c)) { res = res * 10 + (c - '0'); c = get_char(); } return neg ? -res : res; } }; // Helper methods int ri() { return IO::get<int>(); } // }}} #line 10 "Math/tests/matrix_det.test.cpp" #define REP(i, a) for (int i = 0, _##i = (a); i < _##i; ++i) int32_t main() { ios::sync_with_stdio(0); cin.tie(0); int n = IO::get<int>(); Matrix<modint998244353> a(n, n); REP(i,n) REP(j,n) { int x = IO::get<int>(); a[i][j] = x; } auto tmp = a.gauss(); cout << tmp.det().val() << endl; return 0; }