det

Matrix determinant

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Syntax

d = det(A)

Description

d = det(A) returns the determinant of square matrix A.

example

Examples

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Create a 3-by-3 square matrix, A. 

A = [1 -2 4; -5 2 0; 1 0 3]
A = 3×3

     1    -2     4
    -5     2     0
     1     0     3

Calculate the determinant of A. 

d = det(A)
d = 
-32
Open Live Script

Examine why the determinant is not an accurate measure of singularity.

Create a 10-by-10 matrix by multiplying an identity matrix, eye(10), by a small number.

A = eye(10)*0.0001;

The matrix A has very small entries along the main diagonal. However, A is not singular, because it is a multiple of the identity matrix.

Calculate the determinant of A. 

d = det(A)
d = 
1.0000e-40

The determinant is extremely small. A tolerance test of the form abs(det(A)) < tol is likely to flag this matrix as singular. Although the determinant of the matrix is close to zero, A is actually not ill conditioned. Therefore, A is not close to being singular. The determinant of a matrix can be arbitrarily close to zero without conveying information about singularity.

To investigate if A is singular, use either the cond or rcond functions.

Calculate the condition number of A. 

c = cond(A)
c = 
1

The result confirms that A is not ill conditioned.

Open Live Script

Examine a matrix that is exactly singular, but which has a large nonzero determinant. In theory, the determinant of any singular matrix is zero, but because of the nature of floating-point computation, this ideal is not always achievable.

Create a 17-by-17 diagonally dominant singular matrix A and view the pattern of nonzero elements.

A = diag([36 54 24 46 64 78 88 94 96 94 88 78 64 46 24 54 36]);
S = diag([-27 -12 -13 -24 -33 -40 -45 -48 -49 -48 -45 -40 -33 -12 -27 -36],1);
A = A + S + rot90(S,2);
spy(A)

Figure contains an axes object. The axes object with xlabel nz = 49 contains a line object which displays its values using only markers.

A is singular because the rows are linearly dependent. For instance, sum(A) produces a vector of zeros.

Calculate the determinant of A. 

d = det(A)
d = 
2.6281e+10

The determinant of A is quite large despite the fact that A is singular. In fact, the determinant of A should be exactly zero! The inaccuracy of d is due to an aggregation of round-off errors in the MATLAB® implementation of the LU decomposition, which det uses to calculate the determinant. This result demonstrates a few important aspects of calculating numeric determinants. See the Limitations section for more details.

Input Arguments

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Input matrix, specified as a square numeric matrix.

Data Types: single | double
Complex Number Support: Yes

Limitations

Avoid using det to examine if a matrix is singular because of the following limitations. Use cond or rcond instead.

LimitationResult

The magnitude of the determinant is typically unrelated to the condition number of a matrix.

The determinant of a matrix can be arbitrarily large or small without changing the condition number.

det uses the LU decomposition to calculate the determinant, which is susceptible to floating-point round-off errors.

The determinant calculation is sometimes numerically unstable. For example, det can produce a large-magnitude determinant for a singular matrix, even though it should have a magnitude of 0.

Algorithms

det computes the determinant from the triangular factors obtained by Gaussian elimination with the lu function.

[L,U] = lu(X)
s =  det(L)      % This is always +1 or -1 
det(X) = s*prod(diag(U))

Extended Capabilities

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Version History

Introduced before R2006a

See Also

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