Problem:
Score: 300 points

Problem Statement

We have a 3 \times 3 grid. A number c_{i, j} is written in the square (i, j), where (i, j) denotes the square at the i-th row from the top and the j-th column from the left.

According to Takahashi, there are six integers a_1, a_2, a_3, b_1, b_2, b_3 whose values are fixed, and the number written in the square (i, j) is equal to a_i + b_j.

Determine if he is correct.

Constraints

c_{i, j} \ (1 \leq i \leq 3, 1 \leq j \leq 3) is an integer between 0 and 100 (inclusive).

Input

Input is given from Standard Input in the following format:

c_{1,1} c_{1,2} c_{1,3}
c_{2,1} c_{2,2} c_{2,3}
c_{3,1} c_{3,2} c_{3,3}

Output

If Takahashi's statement is correct, print Yes; otherwise, print No.

Sample Input 1

1 0 1
2 1 2
1 0 1

Sample Output 1

Yes

Takahashi is correct, since there are possible sets of integers such as: a_1=0,a_2=1,a_3=0,b_1=1,b_2=0,b_3=1.

Sample Input 2

2 2 2
2 1 2
2 2 2

Sample Output 2

No

Takahashi is incorrect in this case.

Sample Input 3

0 8 8
0 8 8
0 8 8

Sample Output 3

Yes

Sample Input 4

1 8 6
2 9 7
0 7 7

Sample Output 4

No

Solution in C# (Mono 4.6.2.0) (judge verdict: Accepted)

using System;
using System.Collections;
using System.Collections.Generic;
using System.Linq;
using System.Text;

class TEST{
	static void Main(){
		Sol mySol =new Sol();
		mySol.Solve();
	}
}

class Sol{
	public void Solve(){
		double Eps = 1e-9;
		
		double[] v = new double[9];
		for(int i=0;i<3;i++){
			for(int j=0;j<3;j++){
				v[i * 3 + j] = C[i][j];
			}
		}
		double[][] ar = new double[9][];
		for(int i=0;i<9;i++) ar[i] = new double[6];
		for(int i=0;i<3;i++){
			for(int j=0;j<3;j++){
				ar[i * 3 + j][i] = 1;
				ar[i * 3 + j][3 + j] = 1;
			}
		}
		
		var ge = LinearAlg.GaussElimination(ar, v);
		bool chk = true;
		for(int i=0;i<ge.Length;i++){
			bool zeroCoef = true;
			for(int j=0;j<ge[i].Length - 1;j++) if(Math.Abs(ge[i][j]) > Eps) zeroCoef = false;
			if(zeroCoef && Math.Abs(ge[i][ge[i].Length - 1]) > Eps) chk = false;
		}
		Console.WriteLine(chk ? "Yes" : "No");
		
		
	}
	
	
	
	int[][] C;
	public Sol(){
		C = new int[3][];
		for(int i=0;i<3;i++) C[i] = ria();
	}

	static String rs(){return Console.ReadLine();}
	static int ri(){return int.Parse(Console.ReadLine());}
	static long rl(){return long.Parse(Console.ReadLine());}
	static double rd(){return double.Parse(Console.ReadLine());}
	static String[] rsa(char sep=' '){return Console.ReadLine().Split(sep);}
	static int[] ria(char sep=' '){return Array.ConvertAll(Console.ReadLine().Split(sep),e=>int.Parse(e));}
	static long[] rla(char sep=' '){return Array.ConvertAll(Console.ReadLine().Split(sep),e=>long.Parse(e));}
	static double[] rda(char sep=' '){return Array.ConvertAll(Console.ReadLine().Split(sep),e=>double.Parse(e));}
}


class LinearAlg{
	const double Eps = 1e-9;
	
	public static double[][] GaussElimination(double[][] ar,double[] v){
		//  solving  Ax = V in double
		int N = ar.Length;
		int M = ar[0].Length;
		int N0 = Math.Min(N,M);
		
		double[][] L=new double[N][];
		//  returns GaussEliminationMatrix;
		for(int i=0;i<N;i++){
			L[i] = new double[M+1];
			for(int j=0;j<M;j++) L[i][j] = ar[i][j];
			L[i][M] = v[i];
		}
		
		for(int i=0;i<N0;i++){
			int pibot = i;
			double buf = Math.Abs(L[i][i]);
			for(int j=i+1;j<N;j++){
				if(Math.Abs(L[j][i]) > buf + Eps){
					pibot = j;
					buf = Math.Abs(L[j][i]);
				}
			}
			
			if(L[pibot][i] < Eps) continue;
			swap(ref L[i], ref L[pibot]);

			double d = L[i][i];
			for(int j=0;j<=M;j++) L[i][j] /= d;
			
			for(int j=i+1;j<N;j++){
				d = L[j][i];
				for(int k=0;k<=M;k++){
					L[j][k] -= L[i][k] * d;
				}
			}

		}
		
		for(int i=N0-1;i>=0;i--){
			for(int j=i-1;j>=0;j--){
				double d = L[j][i];
				for(int k=0;k<=M;k++){
					L[j][k] -= L[i][k] * d;
				}
			}
		}

		return L;
		
	}
	
	static void swap(ref double[] A,ref double[] B){
		double[] C;
		C=A;A=B;B=C;
	}
}