#include<iostream.h>
class mat
{
	int **matrix;
	int row,col;
public:
	mat(int r,int c);
	void get_element(int a,int b,int value);
	int &disp_element(int a,int b);
	int getrow() {return row;}
	int getcol() {return col;}
	friend mat operator + (mat &m1,mat &m2);
	friend mat operator - (mat &m1,mat &m2);
	~mat()
	{
		delete matrix;
	}
};
mat::mat(int r,int c)
{
	matrix=new int*[r];
	for(int i=0;i<r;i++)
	{
		matrix[i]=new int[c];
	}
}
void mat::get_element(int a,int b,int value)
{
	matrix[a][b]=value;
}
int &mat::disp_element(int a,int b)
{
	 
	return matrix[a][b];    // return encountering access violation during runtime , Help!!! 
}

mat operator + (mat &m1,mat &m2)
{
	int r1=m1.row;
	int c1=m1.col;
	
	mat m(r1,c1);
	for(int i=0;i<r1;i++)
	{
		for(int j=0;j<c1;j++)
		{
			int v=m1.disp_element(i,j)+m2.disp_element(i,j);
			m.get_element(i,j,v);
		}
	}
	return m;
	
}
mat operator - (mat &m1,mat &m2)
{
	int r1=m1.row;
	int c1=m1.col;
	mat m(r1,c1);
	for(int i=0;i<r1;i++)
	{
		for(int j=0;j<c1;j++)
		{
			int v=m1.disp_element(i,j)-m2.disp_element(i,j);
			m.get_element(i,j,v);
		}
	}
	return m;
}

int main()
{
	int x,i,j;
	mat m1(2,2);
	mat m2(2,2);
	cout<<endl<<"Input of m1"<<endl;
	for(i=0;i<2;i++)
	{
		for(j=0;j<2;j++)
		{
			cout<<"enter element row "<<i+1<<" col "<<j+1<<" :";
			cin>>x;
			m1.get_element(i,j,x);
		}
	}
	cout<<endl<<endl<<"display of m1"<<endl;
	for(i=0;i<2;i++)
	{
		for(j=0;j<2;j++)
		{
			cout<<m1.disp_element(i,j)<<"  ";
		}
		cout<<endl;
	}
	//------------------------------------------
	cout<<endl<<"Input of m2"<<endl;
	for(i=0;i<2;i++)
	{
		for(j=0;j<2;j++)
		{
			cout<<"enter element row "<<i+1<<" col "<<j+1<<" :";
			cin>>x;
			m2.get_element(i,j,x);
		}
	}
	cout<<endl<<endl<<"display of m2"<<endl;
	for(i=0;i<2;i++)
	{
		for(j=0;j<2;j++)
		{
			cout<<m2.disp_element(i,j)<<"  ";
		}
		cout<<endl;
	}
	//------------------------------------------
	//Addition and Subtraction
	int r1=m1.getrow();
	int r2=m2.getrow();
	int c1=m1.getcol();
	int c2=m2.getcol();
	mat m3(r1,c1);
	mat m4(r1,c1);
	if((r1!=r2)||(c1!=c2))
	{
		cout<<endl<<"Addition and subtraction not possible.";
	}
	else
	{
			m3=m1+m2;
			m4=m1-m2;
	}

	//------------------------------------------
	
	cout<<endl<<endl<<"display of Sum"<<endl;
	for(i=0;i<2;i++)
	{
		for(j=0;j<2;j++)
		{
			cout<<m3.disp_element(i,j)<<"  ";
		}
		cout<<endl;
	}

	cout<<endl<<endl<<"display of Difference"<<endl;
	for(i=0;i<2;i++)
	{
		for(j=0;j<2;j++)
		{
			cout<<m4.disp_element(i,j)<<"  ";
		}
		cout<<endl;
	}

	return 0;
}

For starters, you aren't properly initializing your row and col data members. Second, you aren't properly copying your objects and end up trying to access freed memory.