## 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