0
sorry i didnt know how to type the matrix out
basically the 5x5 matrix is...
1 2 5 0 1
2 3 7 1 9
1 2 3 0 3
0 0 1 0 0
3 2 -4 0 1
i understand that the 4th row/column is best to delete.
but what do i do from here?
please show step by step if you can :)
Wiki User
∙ 12y ago
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Write a c program to find inverse of a matrix?
#include<stdio.h> #include<math.h> float detrm(float[][],float); void cofact(float[][],float); void trans(float[][],float[][],float); main() { float a[25][25],k,d; int i,j; printf("ENTER THE ORDER OF THE MATRIX:\n"); scanf("%f",&k); printf("ENTER THE ELEMENTS OF THE MATRIX:\n"); for(i=0;i<k;i++) { for(j=0;j<k;j++) { scanf("%f",&a[i][j]); } } d=detrm(a,k); printf("THE DETERMINANT IS=%f",d); if(d==0) printf("\nMATRIX IS NOT INVERSIBLE\n"); else cofact(a,k); } /******************FUNCTION TO FIND THE DETERMINANT OF THE MATRIX************************/ float detrm(float a[25][25],float k) { float s=1,det=0,b[25][25]; int i,j,m,n,c; if(k==1) { return(a[0][0]); } else { det=0; for(c=0;c<k;c++) { m=0; n=0; for(i=0;i<k;i++) { for(j=0;j<k;j++) { b[i][j]=0; if(i!=0&&j!=c) { b[m][n]=a[i][j]; if(n<(k-2)) n++; else { n=0; m++; } } } } det=det+s*(a[0][c]*detrm(b,k-1)); s=-1*s; } } return(det); } /*******************FUNCTION TO FIND COFACTOR*********************************/ void cofact(float num[25][25],float f) { float b[25][25],fac[25][25]; int p,q,m,n,i,j; for(q=0;q<f;q++) { for(p=0;p<f;p++) { m=0; n=0; for(i=0;i<f;i++) { for(j=0;j<f;j++) { b[i][j]=0; if(i!=q&&j!=p) { b[m][n]=num[i][j]; if(n<(f-2)) n++; else { n=0; m++; } } } } fac[q][p]=pow(-1,q+p)*detrm(b,f-1); } } trans(num,fac,f); } /*************FUNCTION TO FIND TRANSPOSE AND INVERSE OF A MATRIX**************************/ void trans(float num[25][25],float fac[25][25],float r) { int i,j; float b[25][25],inv[25][25],d; for(i=0;i<r;i++) { for(j=0;j<r;j++) { b[i][j]=fac[j][i]; } } d=detrm(num,r); inv[i][j]=0; for(i=0;i<r;i++) { for(j=0;j<r;j++) { inv[i][j]=b[i][j]/d; } } printf("\nTHE INVERSE OF THE MATRIX:\n"); for(i=0;i<r;i++) { for(j=0;j<r;j++) { printf("\t%f",inv[i][j]); } printf("\n"); } } ALTERNATIVE CODE: #include<iostream.h> #include<stdio.h> #include<conio.h> #include<process.h> #include<math.h> // Written by Ran // There have been enough comments to help the reader easily understand this program // Helpfulness of COMMENTS in this program and Pre-requisites:- // a. However it's assumed that the reader is familiar with the basics of C++ // b. It is also assumed that the reader knows the basic mathematics involving matrices. // c. Since this program focusses on how to find inverse of a matrix, the comments // in the program are sufficient for understanding this. // It is assumed that the reader is familiar with basics of matrices in C++ (like input, display, // addition,transpose,etc. of matrices) // The comments in this program aim to explain the reader how to find inverse // d. Hence through comments, the reader will also be taught how to find determinant, // adjoint, cofactor,etc. However as said in the previous lines, there won't be comments // for explaining basics like input,display,etc of a matrix using C++. // NOTES: // 1. float datatype has been used for matrix, determinant. // 2. To have consistency between mathematics and C++, this program considers a[1][1] as the first element // i.e row and column indices begin with 1 same as mathematics. // Define a structure matrix with a matrix (2D array of type float) and size n // Declare the objects of this structure used in this program struct matrix { float a[25][25]; int n; }obj,c_obj,trans_obj,obj_cof,obj_adj,obj_inv; // Prototypes of the functions used in this program void input(matrix&); void display(matrix&); matrix reduced(matrix &, int ,int ); float determinant(matrix); float cofactor(matrix,int,int); matrix transpose(matrix); matrix adjoint(matrix); matrix inverse(matrix obj); // Begining of Main function int main() { // Getting dimensions input by the user int r,c; again: cout<<"Enter the order of the matrix: "<<endl; cout<<"Enter Row dimension: "; cin>>r; cout<<"Enter Column dimension: "; cin>>c; // Check dimensions for square matrix so that inverse can be found // If user enters different dimensions for row and column, ask to re-enter or quit program if(r!=c) { char ans; cout<<"Inverse can be found out only for a square matrix. Enter same dimension for row and column. Do you want to enter the dimensions again? Press Y for yes"<<endl; cin>>ans; if(ans=='y') goto again; else cout<<"Program exit"; getch(); exit(0); } // If it's a square matrix, proceed else if(r==c) {obj.n=r;} cout<<endl; input(obj); // call input function to input the matrix elements from the user display(obj); // display the matrix got as input now // Following lines were used to test parts/sections/segments of the code and hence commented /* char ans2; cout<<"do u want to check reduce matrix? Press y to check reduce matrix and press any char to skip this"<<endl; cin>>ans2; if(ans2=='y') { int i,j; cout<<"Enter row i and col j to get reduced matrix"<<endl; cin>>i>>j; // i=i-1; // j=j-1; c_obj=reduced(obj,i,j); char ans1; cout<<"Do you want to display the reduced matrix? If yes, Press y "<<endl; cin>>ans1; if(ans1=='y') { cout<<"Displaying reduced matrix..."<<endl; display(c_obj); } }*/ //Find Determinant cout<<"Finding determinant......"<<endl; cout<<"The determinant is"<<determinant(obj)<<endl; //Find Cofactor if user wishes to char ans3; cout<<"Do you want to find cofactor? Press y if yes"<<endl; cin>>ans3; while(ans3=='y') { int i,j; cout<<"Finding cofactor. Enter row and column"<<endl; cin>>i>>j; cout<<"Cofactor of a["<<i<<"]["<<j<<"] is "<<cofactor(obj,i,j)<<endl; cout<<"want of find cofactor of another element? Press y for yes"<<endl; cin>>ans3; } // Following lines were to meant to test ONLY the transpose function and hence commented /* cout<<"Printing Transpose of the matrix "<<endl; matrix trans1; trans1=transpose(obj); display(trans1); */ // Find Inverse cout<<"\n\n\n Finding Inverse. . .\n\n"; matrix obj_inv2; obj_inv2=inverse(obj); display(obj_inv2); // Display the matrix inverse getch(); return 0; } void input(matrix &obj) { // This function gets elements of a matrix input by the user // Parameter is the structure object obj (used throughout the program) // Parameter is "passed by reference" so as to reflect the changes made by this function, to other functions that call it cout<<"Enter the matrix "<<endl; for (int i=1;i<=obj.n;i++) { for (int j=1;j<=obj.n;j++) { cout<<"Enter the element a["<<i<<"]["<<j<<"] : "; cout<<endl; cin>>obj.a[i][j]; } } } void display(matrix &obj) { // This function displays elements of a matrix passed to it as a parameter // Parameter is the structure object obj (used throughout the program) // Parameter is "passed by reference" but may be "passed by value" also. if(obj.n==0) return; else{ cout<<"The matrix is: "<<endl; for (int i=1;i<=obj.n;i++) { for (int j=1;j<=obj.n;j++) { cout<<obj.a[i][j]<<" "; } cout<<endl; } }} matrix reduced(matrix &obj, int i,int j) { // This function reduces the matrix passed as input to it // The 'reduction' requirement is like this: // Eliminate the row i and column j from the given matrix to get the reduced matrix // This is done by the following logic: // a is given matrix. c_obj is desired reduced matrix // i. Using two for loops (iterating with p and q here) as usual, we scan the given matrix. // row and col represent the current location pointer of row and column of the required reduced matrix. // ii. All elements from given matrix are copied to reduced matrix except for those corresponding to // row i and column j // iii. The reduced matrix has its dimensions one less than that of given matrix int row=1,col=1; for(int p=1;p<=obj.n;p++) // outer loop traverses through rows as usual { for(int q=1;q<=obj.n;q++) // inner loop traverses through columns as usual { if((p!=i)&&(q!=j)) // Skip the elements corresponding to row i OR column j of the given matrix { c_obj.a[row][col]=obj.a[p][q]; col=col+1; } if(col>=obj.n) // When column 'col' of reduced matrix reaches (or exceeds n), reset it to 1 { // and increment 'row'. This means current row of reduced matrix got filled and // we need to begin filling a new row. col=1; row=row+1; if (row>=obj.n) //This represents the case when both 'col' and 'row of reduce matrix reach (or // exceed) n. This means the reduced matrix has been filled up.Break out of the loops. break; } } } c_obj.n=obj.n-1; // Fix the dimension of the reduced matrix one less than the given input matrix return c_obj; // Return the reduced matrix to the calling function. } float determinant(matrix obj) { // This function is called recursively until we get dimension = 1 where the only element in the matrix gets returned. float det=0; if(obj.n==1) {return obj.a[1][1]; } else { for(int scan=1;scan<=obj.n;scan++)//Fix the first row and vary the column in this row using for loop iteration variable 'scan' { det=det+obj.a[1][scan]*int(pow(-1,(1+scan)))*determinant(reduced(obj,1,scan)); // det is calculated to be the sum of the following // i. prev value stored in det. // ii. current element in the first row (i.e. a[1][scan]) MULTIPLIED by -1^(i+j) [i is 1 for 1st row and j is nothing but scan here MULTIPLIED by the reduced matrix corresponding to this i (1) and j (scan) // PLEASE UNDERSTAND BY COMPARING THIS WITH THE MATHEMATICAL WAY OF CALCULATING DETERMINANT // - It's computed in a similar way here. } return det; } } float cofactor(matrix obj,int i,int j) { // The computation done here is like this: // If the matrix (passed as paramenter) has dimension = 1, return the only element as cofactor // Else, return determinant of the reduced matrix corresponding to i and j passed with a // multiplication factor -1^(i+j) float cofact; if(obj.n==1) { return obj.a[1][1]; } else { cofact=int(pow(-1,(i+j)))*determinant(reduced(obj,i,j)); } return cofact; } matrix transpose(matrix obj) { // Transpose matrix is the given matrix with its rows and columns interchanged. // Just invert the elements during storing when scanning through the for loops // trans_obj is the transposed matrix, returned by the function. // obj is the input matrix passed to this function. trans_obj.n=obj.n; for(int i=1;i<=obj.n;i++) { for(int j=1;j<=obj.n;j++) { trans_obj.a[i][j]=obj.a[j][i]; } } return trans_obj; } matrix adjoint(matrix obj) { // obj_adj is adjoint matrix and obj_cof is cofactor matrix // both have dimensions n // Cofactor matrix of a given matrix is a matrix whose elements are the cofactors of the respective // elements of the given matrix // Adjoint matrix is transpose of cofactor matrix. Return this obj_adj.n=obj.n; obj_cof.n=obj.n; for(int i=1;i<=obj.n;i++) { for(int j=1;j<=obj.n;j++) { obj_cof.a[i][j]=cofactor(obj,i,j); } } obj_adj=transpose(obj_cof); return obj_adj; } matrix inverse(matrix obj) { // Formula : Inverse of a matrix is = adj(matrix)/its determinant float d=determinant(obj); // First find determinant of the given matrix matrix obj_null; obj_null.n=0; // Display error message if determinant is 0 if(d==0) { cout<<"Inverse can be found only if the determninant of the matrix is non-zero"<<endl; return obj_null; } // Determinant is non-zero - Proceed finding inverse using the above formula else { matrix obj_adj1=adjoint(obj); obj_adj1.n=obj.n; obj_inv.n=obj.n; for(int i=1;i<=obj.n;i++) { for(int j=1;j<=obj.n;j++) { obj_inv.a[i][j]=(obj_adj1.a[i][j])/d; } } return obj_inv; } }
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d = det(X) returns the determinant of the square matrix X. If X contains only integer entries, the result d is also an integer. The determinant is computed from the triangular factors obtained by Gaussian elimination [L,U] = lu(A) s = det(L) % This is always +1 or -1 det(A) = s*prod(diag(U)) The statement A = [1 2 3; 4 5 6; 7 8 9] produces A = 1 2 3 4 5 6 7 8 9 This happens to be a singular matrix, so d = det(A) produces d = 0. Changing A(3,3) with A(3,3) = 0 turns A into a nonsingular matrix. Now d = det(A) produces d = 27. For more info visit http://www.mathworks.com/access/helpdesk/help/techdoc/ref/det.html
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