首先，定义一个矩阵类作为基类，类中需要包含维度值m（行）和n(列)。并在构造函数中动态申请内存空间，实例化m*n的int型二维数组，重载+和-作为矩阵加减法运算符，包含按照m行n列打印出矩阵的show方法，析构函数包含动态释放内存空间。 接下来，定义一个4*4方阵类作为矩阵类的派生类，包含成员方法求行列式的值，包含成员方法左乘矩阵打印出结果，注意规格检查。 最后，定义一个主函数，随机生成5*6矩阵A，5*6矩阵B，5*6矩阵C，计算A+B，A-C；随机生成4*4方阵D，4*4方阵E，计算D*E，计算D和E的行列式，判断D*A的可计算性。

以下是代码实现：

#include <iostream>
#include <cstdlib>
#include <ctime>

using namespace std;

class Matrix {
public:
    Matrix(int row, int col) : m(row), n(col) {
        data = new int*[m];
        for (int i = 0; i < m; i++) {
            data[i] = new int[n];
        }
    }
    virtual ~Matrix() {
        for (int i = 0; i < m; i++) {
            delete[] data[i];
        }
        delete[] data;
    }
    Matrix operator+(const Matrix& other) const {
        Matrix res(m, n);
        for (int i = 0; i < m; i++) {
            for (int j = 0; j < n; j++) {
                res.data[i][j] = data[i][j] + other.data[i][j];
            }
        }
        return res;
    }
    Matrix operator-(const Matrix& other) const {
        Matrix res(m, n);
        for (int i = 0; i < m; i++) {
            for (int j = 0; j < n; j++) {
                res.data[i][j] = data[i][j] - other.data[i][j];
            }
        }
        return res;
    }
    void show() const {
        for (int i = 0; i < m; i++) {
            for (int j = 0; j < n; j++) {
                cout << data[i][j] << " ";
            }
            cout << endl;
        }
    }
protected:
    int m;
    int n;
    int** data;
};

class SquareMatrix : public Matrix {
public:
    SquareMatrix() : Matrix(4, 4) {}
    int det() const {
        if (m != n) {
            cerr << "Not a square matrix!" << endl;
            return 0;
        }
        if (m == 1) {
            return data[0][0];
        }
        int res = 0;
        for (int i = 0; i < m; i++) {
            SquareMatrix tmp(m - 1, m - 1);
            for (int j = 1; j < m; j++) {
                for (int k = 0; k < m; k++) {
                    if (k < i) {
                        tmp.data[j - 1][k] = data[j][k];
                    } else if (k > i) {
                        tmp.data[j - 1][k - 1] = data[j][k];
                    }
                }
            }
            res += data[0][i] * (i % 2 == 0 ? 1 : -1) * tmp.det();
        }
        return res;
    }
    void leftMultiply(const Matrix& other) const {
        if (n != other.m) {
            cerr << "Invalid matrix size for multiplication!" << endl;
            return;
        }
        SquareMatrix tmp(4, 4);
        for (int i = 0; i < m; i++) {
            for (int j = 0; j < other.n; j++) {
                tmp.data[i][j] = 0;
                for (int k = 0; k < n; k++) {
                    tmp.data[i][j] += data[i][k] * other.data[k][j];
                }
            }
        }
        tmp.show();
    }
};

int main() {
    srand(time(nullptr));

    Matrix A(5, 6), B(5, 6), C(5, 6);
    for (int i = 0; i < A.m; i++) {
        for (int j = 0; j < A.n; j++) {
            A.data[i][j] = rand() % 10;
            B.data[i][j] = rand() % 10;
        }
    }
    cout << "Matrix A:" << endl;
    A.show();
    cout << "Matrix B:" << endl;
    B.show();
    C = A + B;
    cout << "A + B:" << endl;
    C.show();
    C = A - B;
    cout << "A - B:" << endl;
    C.show();

    SquareMatrix D, E;
    for (int i = 0; i < D.m; i++) {
        for (int j = 0; j < D.n; j++) {
            D.data[i][j] = rand() % 10;
            E.data[i][j] = rand() % 10;
        }
    }
    cout << "Matrix D:" << endl;
    D.show();
    cout << "Matrix E:" << endl;
    E.show();
    cout << "D * E:" << endl;
    D.leftMultiply(E);
    cout << "det(D): " << D.det() << endl;
    cout << "det(E): " << E.det() << endl;
    cout << "D * A:" << endl;
    D.leftMultiply(A);

    return 0;
}

以上代码中，Matrix类是矩阵类的基类，实现了矩阵的加减法和打印方法。SquareMatrix类是矩阵类的派生类，继承了Matrix类的成员和方法，并实现了求行列式和左乘矩阵的方法。在主函数中，随机生成两个5*6矩阵A和B，计算A+B和A-B；随机生成两个4*4方阵D和E，计算D*E，计算D和E的行列式，判断D*A的可计算性。