首先，我们需要实现一个二维矩阵类，成员变量为double类型，用来存储矩阵中的元素。然后，我们需要重载赋值运算符、拷贝构造函数和构造函数，实现矩阵的转置操作、矩阵的逆求解、行列式运算、加减乘操作。最后，我们需要实现AX=b求解操作、求矩阵的秩、求矩阵的特征值和特征向量。

以下是实现这个二维矩阵类的代码：

```cpp
#include
using namespace std;

class Matrix {
public:
double mat;
int rows, cols;

Matrix(int r, int c) : rows(r), cols(c) {
mat = new double[rows];
for (int i = 0; i < rows; i++) {
mat[i] = new double[cols];
}
}

~Matrix() {
for (int i = 0; i < rows; i++) {
delete[] mat[i];
}
delete[] mat;
}

void set(int i, int j, double value) {
mat[i][j] = value;
}

Matrix operator=(const Matrix& other) {
if (rows != other.rows || cols != other.cols) {
return this;
}

for (int i = 0; i < rows; i++) {
for (int j = 0; j < cols; j++) {
mat[i][j] = other.mat[i][j];
}
}

return this;
}

Matrix transpose() {
Matrix result(cols, rows);
for (int i = 0; i < rows; i++) {
for (int j = 0; j < cols; j++) {
result.set(j, i, mat[i][j]);
}
}
return result;
}

Matrix inverse() {
if (cols == 1) {
return this;
}

double det = matrix_determinant(mat);
if (det == 0) {
throw "Matrix is singular";
}

Matrix result(cols, cols);
result.mat[0][0] = 1 / det;
for (int i = 1; i < cols; i++) {
result.set(0, i, mat[0][i] result.transpose());
}
return result;
}

void add(Matrix other) {
for (int i = 0; i < rows; i++) {
for (int j = 0; j < cols; j++) {
mat[i][j] += other.mat[i][j];
}
}
}

void subtract(Matrix other) {
for (int i = 0; i < rows; i++) {
for (int j = 0; j < cols; j++) {
mat[i][j] -= other.mat[i][j];
}
}
}

void multiply(Matrix other) {
for (int i = 0; i < rows; i++) {
for (int j = 0; j < other.cols; j++) {
for (int k = 0; k < other.rows; k++) {
mat[i][j] += other.mat[i][k] mat[k][j];
}
}
}
}

void set(int i, int j, double value) {
mat[i][j] = value;
}

double determinant() const {
double det = 0;
for (int i = 0; i < rows; i++) {
det += mat[i][i] matrix_determinant(mat);
}
return det;
}

void print() const {
for (int i = 0; i < rows; i++) {
for (int j = 0; j < cols; j++) {
cout