Dynamic arrays inside a class

I am doing the following exercise at the moment:

A generic Matrix class (15 pt)

a) Create a class called Matrix, it should contain storage for M*N numbers of type double. Just like earlier, when choosing how to store your data it is often useful to know what we are going to use the data for later. In matrix operations we are going to access the different elements of the matrix based on their column and/or row, therefore it is useful to order the members of Matrix as an array. Also, we are going to need to change the size of the data stored in the matrix, therefore it should be dynamically allocated.

b) Create constructors for the matrix.

Create the following three constructors: Matrix() • Default constructor, should initialize the matrix into the invalid state.

explicit Matrix(unsigned int N) • Should construct a valid NxN Matrix, initialized as an identity matrix. (The explicit keyword is not in the syllabus, but it should be used here.)

Matrix(unsigned int M, unsigned int N) • Should construct a valid MxN Matrix, initialized as a Zero matrix. (All elements are zero.)

~Matrix() • The destructor of Matrix, should delete any dynamically allocated memory.

My class this far is as follows:

    class Matrix{
    private:
        int rows;
        int columns;
        double* matrix;
    public:
        Matrix();
        explicit Matrix(int N);
        Matrix(int M, int N);
        ~Matrix();
};

And the rest of my code:

    Matrix::Matrix(){
    double * matrix = NULL;
}

Matrix::Matrix(int N){
    double * matrix = new double[N * N];
    this->rows = N;
    this->columns = N;

    for(int i = 0; i < N; i++){
        for(int j = 0; j < N; j++){
            if(i==j)
                matrix[i * N + j] = 1;
            else
                matrix[i * N + j] = 0;
        }
    }
}

Matrix::Matrix(int M, int N){
    double * matrix = new double[M * N];
    this->rows = M;
    this->columns = N;

    for(int i = 0; i < M; i++){
        for(int j = 0; j < N; j++)
            matrix[i * N 
                                                

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