verify/math/matrix/matrix_rank.test.cpp

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• Last update: 2026-10-06 01:57:24+09:00
• Problem: https://judge.yosupo.jp/problem/matrix_rank

Depends on

• Matrix Linear Algebra (math/matrix/linear_algebra.hpp)
• Dense Matrix (math/matrix/matrix.hpp)
• ModInt (math/modint.hpp)
• Fast IO (utilities/fast_io.hpp)

Code

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

#include "../../../math/matrix/linear_algebra.hpp"
#include "../../../math/modint.hpp"

#include "../../../utilities/fast_io.hpp"

int main() {
    m1une::utilities::FastInput fast_input;
    m1une::utilities::FastOutput fast_output;

    using mint = m1une::math::modint998244353;
    int row_count, column_count;
    fast_input >> row_count >> column_count;
    m1une::matrix::Matrix<mint> matrix(row_count, column_count);
    for (int row = 0; row < row_count; row++) {
        for (int col = 0; col < column_count; col++) {
            fast_input >> matrix[row][col];
        }
    }
    fast_output << m1une::matrix::matrix_rank(std::move(matrix)) << '\n';
}

#line 1 "verify/math/matrix/matrix_rank.test.cpp"
#define PROBLEM "https://judge.yosupo.jp/problem/matrix_rank"

#line 1 "math/matrix/linear_algebra.hpp"



#include <optional>
#include <type_traits>
#include <vector>

#line 1 "math/matrix/matrix.hpp"



#include <cassert>
#include <cstddef>
#include <cstdint>
#include <utility>
#line 9 "math/matrix/matrix.hpp"

namespace m1une {
namespace matrix {

template <class T>
class Matrix {
   private:
    int _rows;
    int _cols;
    std::vector<T> _data;

    static std::size_t storage_size(int rows, int cols) {
        assert(rows >= 0);
        assert(cols >= 0);
        return std::size_t(rows) * std::size_t(cols);
    }

   public:
    using value_type = T;

    Matrix() : _rows(0), _cols(0) {}

    Matrix(int rows, int cols, const T& value = T())
        : _rows(rows), _cols(cols), _data(storage_size(rows, cols), value) {}

    Matrix(int rows, int cols, std::vector<T> values)
        : _rows(rows), _cols(cols), _data(std::move(values)) {
        assert(rows >= 0);
        assert(cols >= 0);
        assert(_data.size() == std::size_t(rows) * std::size_t(cols));
    }

    explicit Matrix(const std::vector<std::vector<T>>& values)
        : _rows(int(values.size())), _cols(values.empty() ? 0 : int(values[0].size())),
          _data(storage_size(_rows, _cols)) {
        for (int row = 0; row < _rows; row++) {
            assert(int(values[std::size_t(row)].size()) == _cols);
            for (int col = 0; col < _cols; col++) {
                (*this)[row][col] = values[std::size_t(row)][std::size_t(col)];
            }
        }
    }

    int rows() const {
        return _rows;
    }

    int cols() const {
        return _cols;
    }

    bool empty() const {
        return _rows == 0 || _cols == 0;
    }

    std::vector<T>& data() {
        return _data;
    }

    const std::vector<T>& data() const {
        return _data;
    }

    T* operator[](int row) {
        assert(0 <= row && row < _rows);
        return _data.data() + std::size_t(row) * std::size_t(_cols);
    }

    const T* operator[](int row) const {
        assert(0 <= row && row < _rows);
        return _data.data() + std::size_t(row) * std::size_t(_cols);
    }

    T& operator()(int row, int col) {
        assert(0 <= col && col < _cols);
        return (*this)[row][col];
    }

    const T& operator()(int row, int col) const {
        assert(0 <= col && col < _cols);
        return (*this)[row][col];
    }

    static Matrix identity(int size) {
        assert(size >= 0);
        Matrix result(size, size);
        for (int i = 0; i < size; i++) result[i][i] = T(1);
        return result;
    }

    Matrix transposed() const {
        Matrix result(_cols, _rows);
        for (int row = 0; row < _rows; row++) {
            for (int col = 0; col < _cols; col++) {
                result[col][row] = (*this)[row][col];
            }
        }
        return result;
    }

    void swap_rows(int first, int second) {
        assert(0 <= first && first < _rows);
        assert(0 <= second && second < _rows);
        if (first == second) return;
        for (int col = 0; col < _cols; col++) {
            std::swap((*this)[first][col], (*this)[second][col]);
        }
    }

    Matrix& operator+=(const Matrix& rhs) {
        assert(_rows == rhs._rows && _cols == rhs._cols);
        for (std::size_t i = 0; i < _data.size(); i++) _data[i] += rhs._data[i];
        return *this;
    }

    Matrix& operator-=(const Matrix& rhs) {
        assert(_rows == rhs._rows && _cols == rhs._cols);
        for (std::size_t i = 0; i < _data.size(); i++) _data[i] -= rhs._data[i];
        return *this;
    }

    Matrix& operator*=(const T& scalar) {
        for (T& value : _data) value *= scalar;
        return *this;
    }

    Matrix& operator/=(const T& scalar) {
        for (T& value : _data) value /= scalar;
        return *this;
    }

    Matrix& operator*=(const Matrix& rhs) {
        return *this = *this * rhs;
    }

    Matrix operator+() const {
        return *this;
    }

    Matrix operator-() const {
        Matrix result = *this;
        for (T& value : result._data) value = T() - value;
        return result;
    }

    friend Matrix operator+(Matrix lhs, const Matrix& rhs) {
        return lhs += rhs;
    }

    friend Matrix operator-(Matrix lhs, const Matrix& rhs) {
        return lhs -= rhs;
    }

    friend Matrix operator*(Matrix lhs, const T& rhs) {
        return lhs *= rhs;
    }

    friend Matrix operator*(const T& lhs, Matrix rhs) {
        return rhs *= lhs;
    }

    friend Matrix operator/(Matrix lhs, const T& rhs) {
        return lhs /= rhs;
    }

    friend Matrix operator*(const Matrix& lhs, const Matrix& rhs) {
        assert(lhs._cols == rhs._rows);
        Matrix result(lhs._rows, rhs._cols);
        for (int row = 0; row < lhs._rows; row++) {
            T* output = result[row];
            for (int middle = 0; middle < lhs._cols; middle++) {
                const T coefficient = lhs[row][middle];
                if (coefficient == T()) continue;
                const T* input = rhs[middle];
                for (int col = 0; col < rhs._cols; col++) {
                    output[col] += coefficient * input[col];
                }
            }
        }
        return result;
    }

    friend std::vector<T> operator*(const Matrix& lhs, const std::vector<T>& rhs) {
        assert(lhs._cols == int(rhs.size()));
        std::vector<T> result(std::size_t(lhs._rows));
        for (int row = 0; row < lhs._rows; row++) {
            T value = T();
            for (int col = 0; col < lhs._cols; col++) {
                value += lhs[row][col] * rhs[std::size_t(col)];
            }
            result[std::size_t(row)] = value;
        }
        return result;
    }

    friend std::vector<T> operator*(const std::vector<T>& lhs, const Matrix& rhs) {
        assert(int(lhs.size()) == rhs._rows);
        std::vector<T> result(std::size_t(rhs._cols));
        for (int row = 0; row < rhs._rows; row++) {
            if (lhs[std::size_t(row)] == T()) continue;
            for (int col = 0; col < rhs._cols; col++) {
                result[std::size_t(col)] += lhs[std::size_t(row)] * rhs[row][col];
            }
        }
        return result;
    }

    bool operator==(const Matrix& rhs) const {
        return _rows == rhs._rows && _cols == rhs._cols && _data == rhs._data;
    }

    bool operator!=(const Matrix& rhs) const {
        return !(*this == rhs);
    }

    Matrix pow(std::uint64_t exponent) const {
        assert(_rows == _cols);
        Matrix result = identity(_rows);
        Matrix base = *this;
        while (exponent > 0) {
            if (exponent & 1) result *= base;
            exponent >>= 1;
            if (exponent > 0) base *= base;
        }
        return result;
    }
};

}  // namespace matrix
}  // namespace m1une


#line 9 "math/matrix/linear_algebra.hpp"

namespace m1une {
namespace matrix {

template <class T>
constexpr T default_epsilon() {
    if constexpr (std::is_floating_point_v<T>) {
        return T(1e-10);
    } else {
        return T();
    }
}

namespace detail {

template <class T>
T matrix_abs(T value) {
    return value < T() ? T() - value : value;
}

template <class T>
bool is_zero(const T& value, const T& eps) {
    if constexpr (std::is_floating_point_v<T>) {
        return matrix_abs(value) <= eps;
    } else {
        (void)eps;
        return value == T();
    }
}

template <class T>
int choose_pivot(const Matrix<T>& matrix, int first_row, int col, const T& eps) {
    int pivot = -1;
    if constexpr (std::is_floating_point_v<T>) {
        for (int row = first_row; row < matrix.rows(); row++) {
            if (is_zero(matrix[row][col], eps)) continue;
            if (pivot == -1 || matrix_abs(matrix[pivot][col]) < matrix_abs(matrix[row][col])) {
                pivot = row;
            }
        }
    } else {
        for (int row = first_row; row < matrix.rows(); row++) {
            if (!is_zero(matrix[row][col], eps)) {
                pivot = row;
                break;
            }
        }
    }
    return pivot;
}

template <class T>
std::vector<int> row_reduce(Matrix<T>& matrix, int pivot_col_limit, const T& eps,
                            bool reduced) {
    std::vector<int> pivot_columns;
    int pivot_row = 0;
    for (int col = 0; col < pivot_col_limit && pivot_row < matrix.rows(); col++) {
        int pivot = choose_pivot(matrix, pivot_row, col, eps);
        if (pivot == -1) continue;
        matrix.swap_rows(pivot_row, pivot);

        const T pivot_value = matrix[pivot_row][col];
        if (reduced) {
            for (int j = col; j < matrix.cols(); j++) matrix[pivot_row][j] /= pivot_value;
        }

        const int first_row = reduced ? 0 : pivot_row + 1;
        for (int row = first_row; row < matrix.rows(); row++) {
            if (row == pivot_row || is_zero(matrix[row][col], eps)) continue;
            T factor = matrix[row][col];
            if (!reduced) factor /= pivot_value;
            matrix[row][col] = T();
            for (int j = col + 1; j < matrix.cols(); j++) {
                matrix[row][j] -= factor * matrix[pivot_row][j];
            }
        }

        pivot_columns.push_back(col);
        pivot_row++;
    }

    if constexpr (std::is_floating_point_v<T>) {
        for (T& value : matrix.data()) {
            if (is_zero(value, eps)) value = T();
        }
    }
    return pivot_columns;
}

}  // namespace detail

template <class T>
struct RowReduction {
    Matrix<T> matrix;
    std::vector<int> pivot_columns;

    int rank() const {
        return int(pivot_columns.size());
    }
};

template <class T>
RowReduction<T> reduced_row_echelon_form(Matrix<T> matrix,
                                         T eps = default_epsilon<T>()) {
    RowReduction<T> result;
    result.pivot_columns = detail::row_reduce(matrix, matrix.cols(), eps, true);
    result.matrix = std::move(matrix);
    return result;
}

template <class T>
int matrix_rank(Matrix<T> matrix, T eps = default_epsilon<T>()) {
    return int(detail::row_reduce(matrix, matrix.cols(), eps, false).size());
}

template <class T>
T determinant(Matrix<T> matrix, T eps = default_epsilon<T>()) {
    assert(matrix.rows() == matrix.cols());
    const int size = matrix.rows();
    T result = T(1);
    bool negate = false;

    for (int col = 0; col < size; col++) {
        int pivot = detail::choose_pivot(matrix, col, col, eps);
        if (pivot == -1) return T();
        if (pivot != col) {
            matrix.swap_rows(pivot, col);
            negate = !negate;
        }

        const T pivot_value = matrix[col][col];
        result *= pivot_value;
        for (int row = col + 1; row < size; row++) {
            if (detail::is_zero(matrix[row][col], eps)) continue;
            const T factor = matrix[row][col] / pivot_value;
            matrix[row][col] = T();
            for (int j = col + 1; j < size; j++) {
                matrix[row][j] -= factor * matrix[col][j];
            }
        }
    }
    return negate ? T() - result : result;
}

template <class T>
std::optional<Matrix<T>> inverse(const Matrix<T>& matrix,
                                 T eps = default_epsilon<T>()) {
    assert(matrix.rows() == matrix.cols());
    const int size = matrix.rows();
    Matrix<T> augmented(size, size * 2);
    for (int row = 0; row < size; row++) {
        for (int col = 0; col < size; col++) {
            augmented[row][col] = matrix[row][col];
        }
        augmented[row][size + row] = T(1);
    }

    const std::vector<int> pivots = detail::row_reduce(augmented, size, eps, true);
    if (int(pivots.size()) != size) return std::nullopt;

    Matrix<T> result(size, size);
    for (int row = 0; row < size; row++) {
        for (int col = 0; col < size; col++) {
            result[row][col] = augmented[row][size + col];
        }
    }
    return result;
}

template <class T>
struct LinearSystemResult {
    bool consistent = false;
    std::vector<T> particular_solution;
    std::vector<std::vector<T>> nullspace_basis;
    std::vector<int> pivot_columns;

    int rank() const {
        return int(pivot_columns.size());
    }

    int nullity() const {
        return consistent ? int(nullspace_basis.size()) : 0;
    }

    bool has_unique_solution() const {
        return consistent && nullspace_basis.empty();
    }
};

template <class T>
LinearSystemResult<T> solve_linear_system(const Matrix<T>& coefficients,
                                          const std::vector<T>& constants,
                                          T eps = default_epsilon<T>()) {
    assert(coefficients.rows() == int(constants.size()));
    const int equation_count = coefficients.rows();
    const int variable_count = coefficients.cols();
    Matrix<T> augmented(equation_count, variable_count + 1);
    for (int row = 0; row < equation_count; row++) {
        for (int col = 0; col < variable_count; col++) {
            augmented[row][col] = coefficients[row][col];
        }
        augmented[row][variable_count] = constants[std::size_t(row)];
    }

    LinearSystemResult<T> result;
    result.pivot_columns =
        detail::row_reduce(augmented, variable_count, eps, true);

    for (int row = result.rank(); row < equation_count; row++) {
        bool zero_left = true;
        for (int col = 0; col < variable_count; col++) {
            if (!detail::is_zero(augmented[row][col], eps)) {
                zero_left = false;
                break;
            }
        }
        if (zero_left && !detail::is_zero(augmented[row][variable_count], eps)) {
            return result;
        }
    }

    result.consistent = true;
    result.particular_solution.assign(std::size_t(variable_count), T());
    std::vector<bool> is_pivot(std::size_t(variable_count), false);
    for (int row = 0; row < result.rank(); row++) {
        const int col = result.pivot_columns[std::size_t(row)];
        is_pivot[std::size_t(col)] = true;
        result.particular_solution[std::size_t(col)] = augmented[row][variable_count];
    }

    for (int free_col = 0; free_col < variable_count; free_col++) {
        if (is_pivot[std::size_t(free_col)]) continue;
        std::vector<T> direction(static_cast<std::size_t>(variable_count));
        direction[std::size_t(free_col)] = T(1);
        for (int row = 0; row < result.rank(); row++) {
            const int pivot_col = result.pivot_columns[std::size_t(row)];
            direction[std::size_t(pivot_col)] = T() - augmented[row][free_col];
        }
        result.nullspace_basis.push_back(std::move(direction));
    }
    return result;
}

}  // namespace matrix
}  // namespace m1une


#line 1 "math/modint.hpp"



#line 6 "math/modint.hpp"
#include <iostream>
#line 9 "math/modint.hpp"

namespace m1une {
namespace math {

template <uint32_t Modulus>
struct ModInt {
    static_assert(0 < Modulus, "Modulus must be positive");

   private:
    uint32_t _v;

   public:
    static constexpr uint32_t mod() {
        return Modulus;
    }

    static constexpr ModInt raw(uint32_t v) noexcept {
        ModInt x;
        x._v = v;
        return x;
    }

    constexpr ModInt() noexcept : _v(0) {}

    template <class Integer, std::enable_if_t<std::is_integral_v<Integer>, int> = 0>
    constexpr ModInt(Integer v) noexcept {
        if constexpr (std::is_signed_v<Integer>) {
            int64_t x = static_cast<int64_t>(v) % static_cast<int64_t>(Modulus);
            if (x < 0) x += Modulus;
            _v = static_cast<uint32_t>(x);
        } else {
            _v = static_cast<uint32_t>(static_cast<uint64_t>(v) % Modulus);
        }
    }

    constexpr uint32_t val() const noexcept {
        return _v;
    }

    constexpr ModInt& operator++() noexcept {
        _v++;
        if (_v == Modulus) _v = 0;
        return *this;
    }

    constexpr ModInt& operator--() noexcept {
        if (_v == 0) _v = Modulus;
        _v--;
        return *this;
    }

    constexpr ModInt operator++(int) noexcept {
        ModInt res = *this;
        ++*this;
        return res;
    }

    constexpr ModInt operator--(int) noexcept {
        ModInt res = *this;
        --*this;
        return res;
    }

    constexpr ModInt& operator+=(const ModInt& rhs) noexcept {
        _v += rhs._v;
        if (_v >= Modulus) _v -= Modulus;
        return *this;
    }

    constexpr ModInt& operator-=(const ModInt& rhs) noexcept {
        _v -= rhs._v;
        if (_v >= Modulus) _v += Modulus;
        return *this;
    }

    constexpr ModInt& operator*=(const ModInt& rhs) noexcept {
        uint64_t z = _v;
        z *= rhs._v;
        _v = static_cast<uint32_t>(z % Modulus);
        return *this;
    }

    constexpr ModInt& operator/=(const ModInt& rhs) noexcept {
        return *this *= rhs.inv();
    }

    constexpr ModInt operator+(const ModInt& rhs) const noexcept {
        return ModInt(*this) += rhs;
    }
    constexpr ModInt operator-(const ModInt& rhs) const noexcept {
        return ModInt(*this) -= rhs;
    }
    constexpr ModInt operator*(const ModInt& rhs) const noexcept {
        return ModInt(*this) *= rhs;
    }
    constexpr ModInt operator/(const ModInt& rhs) const noexcept {
        return ModInt(*this) /= rhs;
    }

    constexpr bool operator==(const ModInt& rhs) const noexcept {
        return _v == rhs._v;
    }
    constexpr bool operator!=(const ModInt& rhs) const noexcept {
        return _v != rhs._v;
    }

    constexpr ModInt pow(long long n) const noexcept {
        ModInt res = raw(1 % Modulus);
        ModInt x = n < 0 ? inv() : *this;
        uint64_t exponent = n < 0 ? uint64_t(-(n + 1)) + 1 : uint64_t(n);
        while (exponent > 0) {
            if (exponent & 1) res *= x;
            x *= x;
            exponent >>= 1;
        }
        return res;
    }

    constexpr ModInt inv() const noexcept {
        int64_t a = _v, b = Modulus, u = 1, v = 0;
        while (b) {
            int64_t t = a / b;
            a -= t * b;
            std::swap(a, b);
            u -= t * v;
            std::swap(u, v);
        }
        assert(a == 1);
        u %= Modulus;
        if (u < 0) u += Modulus;
        return raw(static_cast<uint32_t>(u));
    }

    friend std::ostream& operator<<(std::ostream& os, const ModInt& rhs) {
        return os << rhs._v;
    }

    friend std::istream& operator>>(std::istream& is, ModInt& rhs) {
        long long v;
        is >> v;
        rhs = ModInt(v);
        return is;
    }
};

using modint998244353 = ModInt<998244353>;
using modint1000000007 = ModInt<1000000007>;

template <int Id = 0>
struct DynamicModInt {
   private:
    uint32_t _v;
    inline static uint32_t _mod = 1;

   public:
    static uint32_t mod() noexcept {
        return _mod;
    }

    static void set_mod(uint32_t modulus) noexcept {
        assert(modulus > 0);
        assert(modulus <= uint32_t(1) << 31);
        _mod = modulus;
    }

    static DynamicModInt raw(uint32_t v) noexcept {
        assert(v < _mod);
        DynamicModInt x;
        x._v = v;
        return x;
    }

    DynamicModInt() noexcept : _v(0) {}

    template <class Integer, std::enable_if_t<std::is_integral_v<Integer>, int> = 0>
    DynamicModInt(Integer v) noexcept {
        if constexpr (std::is_signed_v<Integer>) {
            int64_t x = static_cast<int64_t>(v) % static_cast<int64_t>(_mod);
            if (x < 0) x += _mod;
            _v = static_cast<uint32_t>(x);
        } else {
            _v = static_cast<uint32_t>(static_cast<uint64_t>(v) % _mod);
        }
    }

    uint32_t val() const noexcept {
        return _v;
    }

    DynamicModInt& operator++() noexcept {
        _v++;
        if (_v == _mod) _v = 0;
        return *this;
    }

    DynamicModInt& operator--() noexcept {
        if (_v == 0) _v = _mod;
        _v--;
        return *this;
    }

    DynamicModInt operator++(int) noexcept {
        DynamicModInt result = *this;
        ++*this;
        return result;
    }

    DynamicModInt operator--(int) noexcept {
        DynamicModInt result = *this;
        --*this;
        return result;
    }

    DynamicModInt& operator+=(const DynamicModInt& rhs) noexcept {
        _v += rhs._v;
        if (_v >= _mod) _v -= _mod;
        return *this;
    }

    DynamicModInt& operator-=(const DynamicModInt& rhs) noexcept {
        _v -= rhs._v;
        if (_v >= _mod) _v += _mod;
        return *this;
    }

    DynamicModInt& operator*=(const DynamicModInt& rhs) noexcept {
        _v = static_cast<uint32_t>(uint64_t(_v) * rhs._v % _mod);
        return *this;
    }

    DynamicModInt& operator/=(const DynamicModInt& rhs) noexcept {
        return *this *= rhs.inv();
    }

    DynamicModInt operator+(const DynamicModInt& rhs) const noexcept {
        return DynamicModInt(*this) += rhs;
    }

    DynamicModInt operator-(const DynamicModInt& rhs) const noexcept {
        return DynamicModInt(*this) -= rhs;
    }

    DynamicModInt operator*(const DynamicModInt& rhs) const noexcept {
        return DynamicModInt(*this) *= rhs;
    }

    DynamicModInt operator/(const DynamicModInt& rhs) const noexcept {
        return DynamicModInt(*this) /= rhs;
    }

    bool operator==(const DynamicModInt& rhs) const noexcept {
        return _v == rhs._v;
    }

    bool operator!=(const DynamicModInt& rhs) const noexcept {
        return _v != rhs._v;
    }

    DynamicModInt pow(long long exponent) const noexcept {
        DynamicModInt result = raw(1 % _mod);
        DynamicModInt base = exponent < 0 ? inv() : *this;
        uint64_t magnitude =
            exponent < 0 ? uint64_t(-(exponent + 1)) + 1 : uint64_t(exponent);
        while (magnitude > 0) {
            if (magnitude & 1) result *= base;
            base *= base;
            magnitude >>= 1;
        }
        return result;
    }

    DynamicModInt inv() const noexcept {
        int64_t a = _v, b = _mod, u = 1, v = 0;
        while (b) {
            int64_t quotient = a / b;
            a -= quotient * b;
            std::swap(a, b);
            u -= quotient * v;
            std::swap(u, v);
        }
        assert(a == 1);
        u %= _mod;
        if (u < 0) u += _mod;
        return raw(static_cast<uint32_t>(u));
    }

    friend std::ostream& operator<<(std::ostream& os, const DynamicModInt& rhs) {
        return os << rhs._v;
    }

    friend std::istream& operator>>(std::istream& is, DynamicModInt& rhs) {
        long long value;
        is >> value;
        rhs = DynamicModInt(value);
        return is;
    }
};

}  // namespace math
}  // namespace m1une


#line 5 "verify/math/matrix/matrix_rank.test.cpp"

#line 1 "utilities/fast_io.hpp"



#include <algorithm>
#include <array>
#include <cerrno>
#include <charconv>
#line 9 "utilities/fast_io.hpp"
#include <cstdio>
#include <cstdlib>
#line 12 "utilities/fast_io.hpp"
#include <cstring>
#include <iterator>
#include <string>
#include <sys/stat.h>
#line 18 "utilities/fast_io.hpp"
#include <unistd.h>
#line 20 "utilities/fast_io.hpp"

namespace m1une {
namespace utilities {

struct FastOutput;

namespace internal {

// Shared with the convenience helpers in template.hpp.
inline FastOutput* standard_output_instance = nullptr;

// Detect std::begin(x), std::end(x).
template <class T, class = void>
struct is_range : std::false_type {};

template <class T>
struct is_range<T, std::void_t<
    decltype(std::begin(std::declval<T&>())),
    decltype(std::end(std::declval<T&>()))
>> : std::true_type {};

template <class T>
inline constexpr bool is_range_v = is_range<T>::value;

template <class T>
using range_reference_t = decltype(*std::begin(std::declval<T&>()));

template <class T>
using range_value_t = std::remove_cv_t<std::remove_reference_t<range_reference_t<T>>>;

template <class T, class = void>
struct range_stored_value {
    using type = range_value_t<T>;
};

template <class T>
struct range_stored_value<T, std::void_t<typename std::remove_cv_t<std::remove_reference_t<T>>::value_type>> {
    using type = typename std::remove_cv_t<std::remove_reference_t<T>>::value_type;
};

template <class T>
using range_stored_value_t = typename range_stored_value<T>::type;

// Treat strings and C strings as scalar output objects, not as ranges.
template <class T>
struct is_char_array : std::false_type {};

template <class T, std::size_t N>
struct is_char_array<T[N]>
    : std::bool_constant<std::is_same_v<std::remove_cv_t<T>, char>> {};

template <class T>
struct is_string_like
    : std::bool_constant<
          std::is_same_v<std::decay_t<T>, std::string>
          || std::is_same_v<std::decay_t<T>, const char*>
          || std::is_same_v<std::decay_t<T>, char*>
          || is_char_array<std::remove_reference_t<T>>::value
      > {};

template <class T>
inline constexpr bool is_string_like_v = is_string_like<T>::value;

// ModInt-like type: x.val() is printable, and x can be assigned from long long.
template <class T, class = void>
struct has_val_method : std::false_type {};

template <class T>
struct has_val_method<T, std::void_t<decltype(std::declval<const T&>().val())>>
    : std::true_type {};

template <class T>
inline constexpr bool has_val_method_v = has_val_method<T>::value;

template <class T, class = void>
struct has_static_mod_raw : std::false_type {};

template <class T>
struct has_static_mod_raw<
    T, std::void_t<decltype(T::mod()), decltype(T::raw(std::declval<uint32_t>()))>>
    : std::true_type {};

template <class T>
inline constexpr bool has_static_mod_raw_v = has_static_mod_raw<T>::value;

// libstdc++ before GCC 16 does not classify __int128 as an integral type in
// strict ISO modes such as -std=c++23. Keep the fast-I/O interface independent
// of that implementation detail.
template <class T>
inline constexpr bool is_integral_v =
    std::is_integral_v<T>
    || std::is_same_v<std::remove_cv_t<T>, __int128_t>
    || std::is_same_v<std::remove_cv_t<T>, __uint128_t>;

template <class T>
inline constexpr bool is_signed_v =
    std::is_signed_v<T>
    || std::is_same_v<std::remove_cv_t<T>, __int128_t>;

template <class T>
struct make_unsigned {
    using type = std::make_unsigned_t<T>;
};

template <>
struct make_unsigned<__int128_t> {
    using type = __uint128_t;
};

template <>
struct make_unsigned<__uint128_t> {
    using type = __uint128_t;
};

template <class T>
using make_unsigned_t = typename make_unsigned<std::remove_cv_t<T>>::type;

}  // namespace internal

struct FastInput {
    static constexpr int buffer_size = 1 << 20;

   private:
    std::FILE* _stream;
    char _buffer[buffer_size];
    int _position;
    int _length;
    int _file_descriptor;
    bool _streaming;

    bool refill() {
        _position = 0;
        if (_streaming) {
            ssize_t length;
            do {
                length = ::read(_file_descriptor, _buffer, buffer_size);
            } while (length < 0 && errno == EINTR);
            if (length <= 0) {
                _length = 0;
                return false;
            }
            _length = int(length);
        } else {
            _length = int(std::fread(_buffer, 1, buffer_size, _stream));
        }
        return _length != 0;
    }

    template <class T>
    bool read_integer_from_stream(T& value) {
        if (!skip_spaces()) return false;
        int c = read_char_raw();

        bool negative = false;
        if (c == '-') {
            negative = true;
            c = read_char_raw();
        }

        if constexpr (internal::is_signed_v<T>) {
            T result = 0;
            while ('0' <= c && c <= '9') {
                result = negative ? result * 10 - (c - '0')
                                  : result * 10 + (c - '0');
                c = read_char_raw();
            }
            value = result;
        } else {
            T result = 0;
            while ('0' <= c && c <= '9') {
                result = result * 10 + T(c - '0');
                c = read_char_raw();
            }
            value = negative ? T(0) - result : result;
        }
        return true;
    }

    bool prepare_number() {
        if (_length - _position >= 64) return true;
        const int remaining = _length - _position;
        if (remaining > 0) std::memmove(_buffer, _buffer + _position, remaining);
        const int added = int(std::fread(_buffer + remaining, 1, buffer_size - remaining, _stream));
        _position = 0;
        _length = remaining + added;
        if (_length < buffer_size) _buffer[_length] = '\0';
        return _length != 0;
    }

   public:
    explicit FastInput(std::FILE* stream = stdin)
        : _stream(stream),
          _position(0),
          _length(0),
          _file_descriptor(::fileno(stream)),
          _streaming([&] {
              struct stat status;
              return _file_descriptor >= 0
                     && ::fstat(_file_descriptor, &status) == 0
                     && !S_ISREG(status.st_mode);
          }()) {}

    FastInput(const FastInput&) = delete;
    FastInput& operator=(const FastInput&) = delete;

    int read_char_raw() {
        if (_position == _length && !refill()) return EOF;
        return _buffer[_position++];
    }

    bool skip_spaces() {
        int c = read_char_raw();
        while (c != EOF && c <= ' ') c = read_char_raw();
        if (c == EOF) return false;
        --_position;
        return true;
    }

    bool read(char& value) {
        if (!skip_spaces()) return false;
        value = char(read_char_raw());
        return true;
    }

    bool read(std::string& value) {
        if (!skip_spaces()) return false;
        value.clear();
        while (true) {
            const int begin = _position;
            while (_position < _length &&
                   static_cast<unsigned char>(_buffer[_position]) > ' ') {
                ++_position;
            }
            value.append(_buffer + begin, _position - begin);
            if (_position < _length) {
                ++_position;
                return true;
            }
            if (!refill()) return true;
        }
    }

    bool read(bool& value) {
        int x;
        if (!read(x)) return false;
        value = x != 0;
        return true;
    }

    template <class T>
    std::enable_if_t<
        internal::is_integral_v<T>
            && !std::is_same_v<std::remove_cv_t<T>, bool>
            && !std::is_same_v<std::remove_cv_t<T>, char>,
        bool
    >
    read(T& value) {
        if (_streaming) return read_integer_from_stream(value);
        if (!prepare_number()) return false;
        int c = static_cast<unsigned char>(_buffer[_position++]);
        while (c <= ' ') c = static_cast<unsigned char>(_buffer[_position++]);

        bool negative = false;
        if (c == '-') {
            negative = true;
            c = static_cast<unsigned char>(_buffer[_position++]);
        }

        if constexpr (internal::is_signed_v<T>) {
            T result = 0;
            while ('0' <= c && c <= '9') {
                const int first = c - '0';
                const int second = static_cast<unsigned char>(_buffer[_position]) - '0';
                if (0 <= second && second <= 9) {
                    result = negative ? result * 100 - (first * 10 + second)
                                      : result * 100 + (first * 10 + second);
                    ++_position;
                } else {
                    result = negative ? result * 10 - first : result * 10 + first;
                }
                c = static_cast<unsigned char>(_buffer[_position++]);
            }
            value = result;
        } else {
            T result = 0;
            while ('0' <= c && c <= '9') {
                const unsigned first = unsigned(c - '0');
                const int second = static_cast<unsigned char>(_buffer[_position]) - '0';
                if (0 <= second && second <= 9) {
                    result = result * 100 + T(first * 10 + unsigned(second));
                    ++_position;
                } else {
                    result = result * 10 + T(first);
                }
                c = static_cast<unsigned char>(_buffer[_position++]);
            }
            value = negative ? T(0) - result : result;
        }
        if (_position > _length) _position = _length;
        return true;
    }

    template <class T>
    std::enable_if_t<std::is_floating_point_v<T>, bool>
    read(T& value) {
        if (!skip_spaces()) return false;
        int c = read_char_raw();
        bool negative = false;
        if (c == '-' || c == '+') {
            negative = c == '-';
            c = read_char_raw();
        }

        long double result = 0;
        while ('0' <= c && c <= '9') {
            result = result * 10 + (c - '0');
            c = read_char_raw();
        }
        if (c == '.') {
            long double place = 0.1L;
            c = read_char_raw();
            while ('0' <= c && c <= '9') {
                result += (c - '0') * place;
                place *= 0.1L;
                c = read_char_raw();
            }
        }
        if (c == 'e' || c == 'E') {
            c = read_char_raw();
            bool exponent_negative = false;
            if (c == '-' || c == '+') {
                exponent_negative = c == '-';
                c = read_char_raw();
            }
            int exponent = 0;
            while ('0' <= c && c <= '9') {
                exponent = exponent * 10 + (c - '0');
                c = read_char_raw();
            }
            long double scale = 1;
            long double power = 10;
            while (exponent > 0) {
                if (exponent & 1) scale *= power;
                power *= power;
                exponent >>= 1;
            }
            result = exponent_negative ? result / scale : result * scale;
        }
        value = static_cast<T>(negative ? -result : result);
        return true;
    }

    template <class T>
    std::enable_if_t<
        internal::has_val_method_v<T>
            && !internal::is_integral_v<T>
            && !internal::is_range_v<T>,
        bool
    >
    read(T& value) {
        long long x;
        if (!read(x)) return false;
        if constexpr (internal::has_static_mod_raw_v<T>) {
            if (x >= 0 && uint64_t(x) < uint64_t(T::mod())) {
                value = T::raw(uint32_t(x));
            } else {
                value = T(x);
            }
        } else {
            value = T(x);
        }
        return true;
    }

    template <class First, class Second>
    bool read(std::pair<First, Second>& value) {
        if (!read(value.first)) return false;
        return read(value.second);
    }

    template <class Range>
    std::enable_if_t<
        internal::is_range_v<Range>
            && !internal::is_string_like_v<Range>,
        bool
    >
    read(Range& range) {
        using StoredValue = internal::range_stored_value_t<Range>;
        constexpr bool nested = internal::is_range_v<StoredValue>
                                && !internal::is_string_like_v<StoredValue>;

        for (auto&& value : range) {
            if constexpr (std::is_same_v<StoredValue, bool> && !nested) {
                bool x;
                if (!read(x)) return false;
                value = x;
            } else {
                if (!read(value)) return false;
            }
        }
        return true;
    }

    template <class First, class Second, class... Rest>
    bool read(First& first, Second& second, Rest&... rest) {
        if (!read(first)) return false;
        return read(second, rest...);
    }

    template <class T>
    FastInput& operator>>(T& value) {
        if (!read(value)) std::abort();
        return *this;
    }
};

struct FastOutput {
    static constexpr int buffer_size = 1 << 20;

   private:
    inline static const auto digit_quads = [] {
        std::array<char, 40000> result{};
        for (int i = 0; i < 10000; i++) {
            int value = i;
            for (int j = 3; j >= 0; j--) {
                result[4 * i + j] = char('0' + value % 10);
                value /= 10;
            }
        }
        return result;
    }();

    std::FILE* _stream;
    char _buffer[buffer_size];
    int _position;
    int _precision;
    std::chars_format _float_format;
    char _range_separator;
    std::string* _capture = nullptr;

    template <class T>
    std::string format_cell(const T& value) {
        std::string result;
        struct CaptureGuard {
            std::string*& target;
            std::string* previous;
            ~CaptureGuard() { target = previous; }
        } guard{_capture, _capture};
        _capture = &result;
        write(value);
        return result;
    }

    template <class Matrix>
    void write_aligned_matrix(const Matrix& matrix) {
        std::vector<std::vector<std::string>> rows;
        std::vector<std::size_t> widths;
        for (const auto& row : matrix) {
            auto& cells = rows.emplace_back();
            std::size_t column = 0;
            for (const auto& value : row) {
                cells.push_back(format_cell(value));
                if (column == widths.size()) widths.push_back(0);
                widths[column] = std::max(widths[column], cells.back().size());
                ++column;
            }
        }
        bool first = true;
        for (const auto& row : rows) {
            if (!first) write_char('\n');
            first = false;
            for (std::size_t column = 0; column < row.size(); ++column) {
                if (column != 0) write_char(_range_separator);
                for (std::size_t padding = row[column].size();
                     padding < widths[column]; ++padding) {
                    write_char(' ');
                }
                write(row[column]);
            }
        }
    }

   public:
    explicit FastOutput(std::FILE* stream = stdout)
        : _stream(stream),
          _position(0),
          _precision(6),
          _float_format(std::chars_format::general),
          _range_separator(' ') {
        if (_stream == stdout
            && internal::standard_output_instance == nullptr) {
            internal::standard_output_instance = this;
        }
    }

    FastOutput(const FastOutput&) = delete;
    FastOutput& operator=(const FastOutput&) = delete;

    ~FastOutput() {
        flush();
        if (internal::standard_output_instance == this) {
            internal::standard_output_instance = nullptr;
        }
    }

    void flush() {
        if (_position != 0) {
            std::fwrite(_buffer, 1, _position, _stream);
            _position = 0;
        }
        std::fflush(_stream);
    }

    void write_char(char c) {
        if (_capture != nullptr) {
            _capture->push_back(c);
            return;
        }
        if (_position == buffer_size) flush();
        _buffer[_position++] = c;
    }

    void write(const char* s) {
        while (*s != '\0') write_char(*s++);
    }

    void write(const std::string& s) {
        if (_capture != nullptr) {
            _capture->append(s);
            return;
        }
        std::size_t position = 0;
        while (position < s.size()) {
            if (_position == buffer_size) flush();
            const std::size_t copied =
                std::min<std::size_t>(buffer_size - _position, s.size() - position);
            std::memcpy(_buffer + _position, s.data() + position, copied);
            _position += int(copied);
            position += copied;
        }
    }

    void write(char c) {
        write_char(c);
    }

    void write(bool value) {
        write_char(value ? '1' : '0');
    }

    template <class T>
    std::enable_if_t<std::is_floating_point_v<T>>
    write(T value) {
        char digits[128];
        auto [end, error] = std::to_chars(
            digits,
            digits + sizeof(digits),
            value,
            _float_format,
            _precision
        );
        if (error != std::errc()) std::abort();
        for (const char* pointer = digits; pointer != end; pointer++) {
            write_char(*pointer);
        }
    }

    template <class T>
    std::enable_if_t<
        internal::is_integral_v<T>
            && !std::is_same_v<std::remove_cv_t<T>, bool>
            && !std::is_same_v<std::remove_cv_t<T>, char>
    >
    write(T value) {
        using Raw = std::remove_cv_t<T>;
        using Unsigned = internal::make_unsigned_t<Raw>;

        Unsigned magnitude;
        if constexpr (internal::is_signed_v<Raw>) {
            if (value < 0) {
                write_char('-');
                magnitude = Unsigned(0) - Unsigned(value);
            } else {
                magnitude = Unsigned(value);
            }
        } else {
            magnitude = value;
        }

        if (magnitude == 0) {
            write_char('0');
            return;
        }

        unsigned chunks[16];
        int count = 0;
        while (magnitude >= 10000) {
            const Unsigned quotient = magnitude / 10000;
            chunks[count++] = unsigned(magnitude - quotient * 10000);
            magnitude = quotient;
        }
        if (_capture == nullptr && _position > buffer_size - 64) flush();
        char captured[64];
        char* const begin = _capture != nullptr ? captured : _buffer + _position;
        char* destination = begin;
        const unsigned leading = unsigned(magnitude);
        const char* first = digit_quads.data() + 4 * leading;
        int skip = leading < 10 ? 3 : leading < 100 ? 2 : leading < 1000 ? 1 : 0;
        for (; skip < 4; skip++) *destination++ = first[skip];
        while (count--) {
            const char* digits = digit_quads.data() + 4 * chunks[count];
            std::memcpy(destination, digits, 4);
            destination += 4;
        }
        if (_capture != nullptr) {
            _capture->append(begin, destination - begin);
        } else {
            _position += int(destination - begin);
        }
    }

    template <class T>
    std::enable_if_t<
        internal::has_val_method_v<T>
            && !internal::is_integral_v<T>
            && !internal::is_range_v<T>
    >
    write(const T& value) {
        write(value.val());
    }

    template <class First, class Second>
    void write(const std::pair<First, Second>& value) {
        write(value.first);
        write_char(' ');
        write(value.second);
    }

    template <class Range>
    std::enable_if_t<
        internal::is_range_v<Range>
            && !internal::is_string_like_v<Range>
    >
    write(const Range& range) {
        using StoredValue = internal::range_stored_value_t<const Range>;
        constexpr bool nested = internal::is_range_v<StoredValue>
                                && !internal::is_string_like_v<StoredValue>;

        bool first = true;
        for (const auto& value : range) {
            if (!first) write_char(nested ? '\n' : _range_separator);
            first = false;
            if constexpr (std::is_same_v<StoredValue, bool> && !nested) {
                write(static_cast<bool>(value));
            } else {
                write(value);
            }
        }
    }

    template <class First, class... Rest>
    void print(const First& first, const Rest&... rest) {
        write(first);
        ((write_char(' '), write(rest)), ...);
    }

    void println() {
        write_char('\n');
    }

    void set_precision(int precision) {
        _precision = precision;
    }

    void set_fixed(int precision = 6) {
        _float_format = std::chars_format::fixed;
        _precision = precision;
    }

    void set_general(int precision = 6) {
        _float_format = std::chars_format::general;
        _precision = precision;
    }

    void set_range_separator(char separator) {
        _range_separator = separator;
    }

    template <class Matrix>
    void write_aligned(const Matrix& matrix) {
        using Row = internal::range_stored_value_t<const Matrix>;
        using Cell = internal::range_stored_value_t<const Row>;
        static_assert(internal::is_range_v<Row> && !internal::is_string_like_v<Row>,
                      "write_aligned requires a two-dimensional range");
        static_assert(!internal::is_range_v<Cell> || internal::is_string_like_v<Cell>,
                      "write_aligned requires scalar cells");
        write_aligned_matrix(matrix);
    }

    template <class Matrix>
    void println_aligned(const Matrix& matrix) {
        write_aligned(matrix);
        write_char('\n');
    }

    template <class... Args>
    void println(const Args&... args) {
        print(args...);
        write_char('\n');
    }

    template <class T>
    FastOutput& operator<<(const T& value) {
        write(value);
        return *this;
    }
};

}  // namespace utilities
}  // namespace m1une


#line 7 "verify/math/matrix/matrix_rank.test.cpp"

int main() {
    m1une::utilities::FastInput fast_input;
    m1une::utilities::FastOutput fast_output;

    using mint = m1une::math::modint998244353;
    int row_count, column_count;
    fast_input >> row_count >> column_count;
    m1une::matrix::Matrix<mint> matrix(row_count, column_count);
    for (int row = 0; row < row_count; row++) {
        for (int col = 0; col < column_count; col++) {
            fast_input >> matrix[row][col];
        }
    }
    fast_output << m1une::matrix::matrix_rank(std::move(matrix)) << '\n';
}

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