Determinant of a Square MatrixWritten by Paul BourkeJune 2001 The determinant of an n by n matrix will be written as follows
The solution is given by the so called "determinant expansion by minors". A minor Mij of the matrix A is the n-1 by n-1 matrix made by the rows and columns of A except the i'th row and the j'th column is not included. So for example M12 for the matrix A above is given below
The determinant is the given by the following where the sum is taken over a single row or column.
Any row or column can be chosen across which to take the sum, when computing manually the row or column with the most zeros is chosen to minimise the amount of work. If the first row is chosen for the sum then the determinant in terms of the minors would be
The process is repeated for each of the determinants above, on each expansion the dimension of the determinant in each term decreases by 1. When the terms are 2 by 2 matrices the determinant is given as
If the determinant is 0 the matrix said to be "singular". A singular matrix either has zero elements in an entire row or column, or else a row (or column) is a linear combination of other rows (or columns). C source code example/* 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(-1.0,1.0+j1+1.0) * a[0][j1] * Determinant(m,n-1); for (i=0;i<n-1;i++) free(m[i]); free(m); } } return(det); }Contribution by Edward Popko, a well commented version: determinant.c for Microsoft C++ Visual Studio 6.0.
Inverse of a square matrixWritten by Paul BourkeAugust 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
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 multiplied by its inverse gives the identity matrix (all zeros except along the diagonal which are ones).
Inverse of a 2x2 matrix The inverse of a 2x2 matrix can be written explicitly, namely
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; } } } |