# Determinant of an Arbitrary Square Matrix

Written by Paul Bourke
June 2001

The determinant of an n by n matrix will be written as follows

det(A) = |A| = a11 a12 a13 . . . a1n a21 a22 a23 . . . a2n a31 a32 a33 . . . a3n : : : aij : an1 an2 an3 . . . ann 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

M12 = a21 a23 a24 . . . a2n a31 a33 a34 . . . a3n a41 a43 a44 . . . a4n : : : aij, i!=1, j!=2 : an1 an3 an4 . . . ann The determinant is the given by the following where the sum is taken over a single row or column.

 |A| = (-1)i+j aij Mij

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

|A| = a11 M11 - a12 M12 + a13 M13 - . . . + a1n M1n
Or expanded out as follows.
|A| = a11 a22 a23 a24 . . . a2n a32 a33 a34 . . . a3n a42 a43 a44 . . . a4n : : : . . . : an2 an3 an4 . . . ann - a12 a21 a23 a24 . . . a2n a31 a33 a34 . . . a3n a41 a43 a44 . . . a4n : : : . . . : an1 an3 an4 . . . ann + a13 a21 a22 a24 . . . a2n a31 a32 a34 . . . a3n a41 a42 a44 . . . a4n : : : . . . : an1 an2 an4 . . . ann . . . . . + a1n a21 a22 a23 . . . a2(n-1) a31 a32 a33 . . . a3(n-1) a41 a42 a43 . . . a4(n-1) : : : . . . : an1 an2 an3 . . . an(n-1) The process is repreated 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 a11 a12 a21 a22 = a11 a22 - a12 a21

If the determinant is 0 the matrix said to be "singular". A singular matrix either has izero 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;
} else if (n == 2) {
det = a * a - a * a;
} 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[j1] * Determinant(m,n-1);
for (i=0;i<n-1;i++)
free(m[i]);
free(m);
}
}
return(det);
}
```