A general 2d matrix has two different dimensions, rows and columns (unless it is the particular case of a square matrix).
In the general case, your code could be:
#include <stdexcept>
class matrix
{
public:
    matrix(int rows, int columns) :
        mRows(rows), mColumns(columns), mP(NULL)
    {
        mP = new double[mRows * mColumns];
        for (int i = 0; i < mRows * mColumns; i++)
            mP[i] = 0.0;
    }
    matrix(const matrix& other) :
        mRows(other.mRows), mColumns(other.mColumns), mP(NULL)
    {
        mP = new double[mRows * mColumns];
        for (int i = 0; i < mRows * mColumns; i++)
            mP[i] = other.mP[i];
    }
    ~matrix()
    {
        delete[] mP;
    }
    const double* operator[] (int row) const {
        return &mP[row * mColumns];
    }
    double* operator[] (int row) {
        return &mP[row * mColumns];
    }
    matrix& operator=(const matrix& other) {
        if (this == &other) return *this;
        if ((mRows != other.mRows) || (mColumns != other.mColumns)) throw std::invalid_argument( "dimensions don't match" );
        for (int r = 0; r < mRows; r++) {
            for (int c = 0; c < mColumns; c++) {
                (*this)[r][c] = other[r][c];
            }
        }
        return *this;
    }
    matrix& operator+=(const matrix& other) {
        if ((mRows != other.mRows) || (mColumns != other.mColumns)) throw std::invalid_argument( "dimensions don't match" );
        for (int r = 0; r < mRows; r++) {
            for (int c = 0; c < mColumns; c++) {
                (*this)[r][c] += other[r][c];
            }
        }
        return *this;
    }
private:
    int mRows;
    int mColumns;
    double *mP;
};
inline matrix operator+(matrix lhs, const matrix& rhs)
{
    lhs += rhs;
    return lhs;
}