代数余子式矩阵求行列式

Posted 2020-09-23 z1j1n1

因为在删除一条边时矩阵只有一行上的两个值发生变化，将上述法则代入该行即可。

#include <cstdio>
#include <cmath>
#define LL long long 
using namespace std;

  int n,m;
  LL sid[100001][3];
  LL tot;
  const LL mo=1e9+7;
  
  LL qpow(LL bas,int powe){
      LL ret=1;
      for (;powe;bas*=bas,bas%=mo){
        if (powe&1) ret*=bas,ret%=mo;
      powe=powe>>1;    
    }
    return(ret);
  }
    
  struct matrix{
      LL a[601][601],tmp[601][601];
      int n,m;
      
      void cpy(matrix&b){
        n=b.n;m=b.m;
      for (int i=1;i<=n;i++)    
        for (int j=1;j<=m;j++)
          a[i][j]=b.a[i][j];
    }
      
      void mul(matrix &b){
      for (int i=0;i<=n;i++) 
        for (int j=0;j<=b.m;j++) 
          tmp[i][j]=0;
        
      for (int i=0;i<=n;i++)
        for (int k=0;k<=m;k++)
          if (a[i][k]) 
            for (int j=0;j<=b.m;j++)
              tmp[i][j]+=a[i][k]*b.a[k][j]%mo,tmp[i][j]%=mo;
        
      for (int i=0;i<=n;i++)
        for (int j=0;j<=b.m;j++)
          a[i][j]=tmp[i][j];
      m=b.m;
    }
    
    LL det(){
      for (int i=1;i<=n;i++) for (int j=1;j<=m;j++) tmp[i][j]=a[i][j];
      for (int i=1;i<=n;i++){
        for (int j=i+1;j<=n;j++){
          LL bas=a[j][i]*qpow(a[i][i],mo-2)%mo;
          for (int k=i;k<=n;k++)
            a[j][k]-=a[i][k]*bas%mo,a[j][k]%=mo,a[j][k]+=mo,a[j][k]%=mo;
          }    
      }
      LL ret=1;
      for (int i=1;i<=n;i++) ret*=a[i][i],ret%=mo;
      for (int i=1;i<=n;i++) for (int j=1;j<=m;j++) a[i][j]=tmp[i][j];
      return(ret);
    }
    
    void mul(LL num){
      for (int i=1;i<=n;i++)
        for (int j=1;j<=m;j++)
          a[i][j]*=num,a[i][j]%=mo;
    }
    
    void getinv(){
      for (int i=1;i<=n;i++) a[i][n+i]=1;
      for (int i=1;i<=n;i++) for (int j=1;j<=n;j++) a[i][j]=(a[i][j]%mo+mo)%mo;
      for (int i=1;i<=n;i++){
        int po;LL maxi=0;
        for (int j=i;j<=n;j++){
          if (fabs(a[j][i])>maxi){
            maxi=fabs(a[j][i]);po=j;
          }
        }
        for (int j=1;j<=2*n;j++){
          LL t=a[i][j];a[i][j]=a[po][j];a[po][j]=t;
        }
        if (fabs(maxi)==0) continue;
            
        for (int j=i+1;j<=n;j++){
          LL tim=a[j][i]*qpow(a[i][i],mo-2)%mo;
          for (int k=i;k<=2*n;k++) a[j][k]-=a[i][k]*tim%mo,a[j][k]%=mo;
          }
        }
        
        for (int i=1;i<=n;i++) for (int j=1;j<=2*n;j++) a[i][j]=(a[i][j]%mo+mo)%mo;
        for (int i=n;i>=1;i--){
          for (int j=i+1;j<=n;j++){
            for (int k=n+1;k<=2*n;k++)
              a[i][k]-=a[i][j]*a[j][k]%mo,a[i][k]%=mo;
            a[i][j]=0;          
          }
          for (int j=n+1;j<=2*n;j++)
            a[i][j]*=qpow(a[i][i],mo-2),a[i][j]%=mo;
          a[i][i]=1;  
        }
        
        for (int i=1;i<=n;i++)
          for (int j=1;j<=n;j++)
            a[i][j]=(a[i][j+n]%mo+mo)%mo;
    }
    
    void trav(){
      for (int i=1;i<=n;i++)
        for (int j=1;j<=n;j++)
          tmp[i][j]=a[j][i];
      for (int i=1;i<=n;i++)
        for (int j=1;j<=n;j++)
          a[i][j]=tmp[i][j];
    }
  }a,A,inv;

  int main(){      
      scanf("%d%d",&n,&m);
      a.n=a.m=n-1;
      for (int i=1;i<=m;i++){
        scanf("%lld%lld%lld",&sid[i][0],&sid[i][1],&sid[i][2]);
        a.a[sid[i][0]][sid[i][1]]--;
        a.a[sid[i][0]][sid[i][0]]++;
    }
    A.n=A.m=n-1;
    for (int i=1;i<n;i++) A.a[i][i]=1;
    A.mul(tot=a.det());
    
    inv.cpy(a);
    inv.getinv();
    A.mul(inv);
    A.trav();
    
    LL ans=0;
    for (int i=1;i<=m;i++){
      if (sid[i][0]==n) continue;
      a.a[sid[i][0]][sid[i][1]]++;
        a.a[sid[i][0]][sid[i][0]]--;
      LL ttot=0;
      for (int j=1;j<n;j++)
        ttot+=a.a[sid[i][0]][j]*A.a[sid[i][0]][j]%mo,ttot%=mo;
      ans+=(tot-ttot)*sid[i][2]%mo;ans%=mo; 
      a.a[sid[i][0]][sid[i][1]]--;
        a.a[sid[i][0]][sid[i][0]]++;
    }
    printf("%lld\\n",(ans%mo+mo)%mo);
  }