DET Determinant of a Matrix

Section: Array Generation and Manipulations

Usage

Calculates the determinant of a matrix. Note that for all but very small problems, the determinant is not particularly useful. The condition number cond gives a more reasonable estimate as to the suitability of a matrix for inversion than comparing det(A) to zero. In any case, the syntax for its use is 
  y = det(A)

where A is a square matrix.

Function Internals

The determinant is calculated via the LU decomposition. Note that the determinant of a product of matrices is the product of the determinants. Then, we have that

where L is lower triangular with 1s on the main diagonal, U is upper triangular, and P is a row-permutation matrix. Taking the determinant of both sides yields

where we have used the fact that the determinant of L is 1. The determinant of P (which is a row exchange matrix) is either 1 or -1.

Example

Here we assemble a random matrix and compute its determinant 
--> A = rand(5);
--> det(A)

ans = 
   -0.1160

Then, we exchange two rows of A to demonstrate how the determinant changes sign (but the magnitude is the same) 

--> B = A([2,1,3,4,5],:);
--> det(B)

ans = 
    0.1160