# det

## Purpose

Matrix determinant.

## Synopsis

    d = det(X)

## Description

`d = det(X)` is the determinant of the square matrix `X`. If `X` contains only integer entries, the result `d` is also an integer.

Using `det(X) == 0` as a test for matrix singularity is appropriate only for matrices of modest order with small integer entries. Testing singularity using `abs(det(X)) <= tolerance` is rarely a good idea because it is difficult to choose the correct tolerance. The function `rcond(X)` is intended to check for singular and nearly singular matrices. See `rcond` for details.

## Algorithm

The determinant is computed from the triangular factors obtained by Gaussian elimination

    [L,U] = lu(A)
    s = +1 or -1 = det(L)
    det(A) = s*prod(diag(U))

## Examples

The statement

    A = [1  2  3; 4  5  6; 7  8  9]

produces

    A =
        1     2     3
        4     5     6
        7     8     9

This happens to be a singular matrix, so

    d = det(A)

produces

    d =
        0

Changing `A(3,3)` with

    A(3,3) = 0;

turns `A` into a nonsingular matrix, so now

    d = det(A)

produces

    d =
        27

## See Also

     \, /, inv, lu, rcond, rref

(c) Copyright 1994 by The MathWorks, Inc.