MATLAB Function Reference  det

Matrix determinant

Syntax

• ```d = det(X)
```

Description

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

Remarks

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 not recommended as it is difficult to choose the correct tolerance. The function `cond(X)` can check for singular and nearly singular matrices.

Algorithm

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

• ```[L,U] = lu(A)
s =  det(L)        % This is always +1 or -1
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. Now `d = det(A) `produces `d = 27`.

`cond`, `condest`, `inv`, `lu`, `rref`
The arithmetic operators `\`, `/`