Inverse of a square matrix

Written by Paul Bourke
August 2002

See also: Determinant of a Square Matrix

The inverse of a square matrix A with a non zero determinant is the adjoint matrix divided by the determinant, this can be written as

 
1
  
A-1 =
 adj(A)
 
det(A)
  

The adjoint matrix is the transpose of the cofactor matrix. The cofactor matrix is the matrix of determinants of the minors Aij multiplied by -1i+j. The i,j'th minor of A is the matrix A without the i'th column or the j'th row.

Example (3x3 matrix)

The following example illustrates each matrix type and at 3x3 the steps can be readily calculated on paper. It can also be verified that the original matrix A multipled by its inverse gives the identity matrix (all zeros except along the diagonal which are ones).

A =
  1  
  2  
  0  
  -1  
  1  
  1  
  1  
  2  
  3  

Determinant = 9

Cofactor matrix =   
  
  +
  1  
  1  
  2  
  3  
  
  
  -
  -1  
  1  
  1  
  3  
  
  
  +
  -1  
  1  
  1  
  2  
  
  
  -
  2  
  0  
  2  
  3  
  
  
  +
  1  
  0  
  1  
  3  
  
  
  -
  1  
  2  
  1  
  2  
  
  
  +
  2  
  0  
  1  
  1  
  
  
  -
  1  
  0  
  -1  
  1  
  
  
  +
  1  
  2  
  -1  
  1  
  
       =  
  1  
  4  
  -3  
  -6  
  3  
  -0  
  2  
  -1  
  3  

Adjoint matrix = Transpose of cofactor matrix =
  1  
  -6  
  2  
  4  
  3  
  -1  
  -3  
  -0  
  3  

A-1 = Inverse of A =
  1/9  
  -6/9  
  2/9  
  4/9  
  3/9  
  -1/9  
  -3/9  
  0  
  3/9  

A A-1 =
  1  
  0  
  0  
  0  
  1  
  0  
  0  
  0  
  1  

Inverse of a 2x2 matrix

The inverse of a 2x2 matrix can be written explicitly, namely

  a  
  b  
  c  
  d  
 = 
1
  
ad - bc
  d  
  -b  
  -c  
  a  
Source code

The three functions required are the determinant, cofactor, and transpose. Examples of these are given below. 

/*
   Recursive definition of determinate using expansion by minors.
*/
double Determinant(double **a,int n)
{
   int i,j,j1,j2;
   double det = 0;
   double **m = NULL;

   if (n < 1) { /* Error */

   } else if (n == 1) { /* Shouldn't get used */
      det = a[0][0];
   } else if (n == 2) {
      det = a[0][0] * a[1][1] - a[1][0] * a[0][1];
   } else {
      det = 0;
      for (j1=0;j1<n;j1++) {
         m = malloc((n-1)*sizeof(double *));
         for (i=0;i<n-1;i++)
            m[i] = malloc((n-1)*sizeof(double));
         for (i=1;i<n;i++) {
            j2 = 0;
            for (j=0;j<n;j++) {
               if (j == j1)
                  continue;
               m[i-1][j2] = a[i][j];
               j2++;
            }
         }
         det += pow(-1.0,j1+2.0) * a[0][j1] * Determinant(m,n-1);
         for (i=0;i<n-1;i++)
            free(m[i]);
         free(m);
      }
   }
   return(det);
}

/*
   Find the cofactor matrix of a square matrix
*/
void CoFactor(double **a,int n,double **b)
{
   int i,j,ii,jj,i1,j1;
   double det;
   double **c;

   c = malloc((n-1)*sizeof(double *));
   for (i=0;i<n-1;i++)
     c[i] = malloc((n-1)*sizeof(double));

   for (j=0;j<n;j++) {
      for (i=0;i<n;i++) {

         /* Form the adjoint a_ij */
         i1 = 0;
         for (ii=0;ii<n;ii++) {
            if (ii == i)
               continue;
            j1 = 0;
            for (jj=0;jj<n;jj++) {
               if (jj == j)
                  continue;
               c[i1][j1] = a[ii][jj];
               j1++;
            }
            i1++;
         }

         /* Calculate the determinate */
         det = Determinant(c,n-1);

         /* Fill in the elements of the cofactor */
         b[i][j] = pow(-1.0,i+j+2.0) * det;
      }
   }
   for (i=0;i<n-1;i++)
      free(c[i]);
   free(c);
}

/*
   Transpose of a square matrix, do it in place
*/
void Transpose(double **a,int n)
{
   int i,j;
   double tmp;

   for (i=1;i<n;i++) {
      for (j=0;j<i;j++) {
         tmp = a[i][j];
         a[i][j] = a[j][i];
         a[j][i] = tmp;
      }
   }
}