Det

Det[m]

gives the determinant of the square matrix m.

Details and Options

  • Det[m,Modulus->n] computes the determinant modulo n.

Examples

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Basic Examples  (2)

Find the determinant of a symbolic matrix:

Scope  (3)

Use exact arithmetic to compute the determinant:

Use machine arithmetic:

Use 40-digit precision arithmetic:

Determinant of a complex-valued matrix:

Determinant of a sparse matrix:

Options  (1)

Modulus  (1)

Compute a determinant using arithmetic modulo 47:

This is faster than computing Mod[Det[m],47]:

Applications  (2)

Cramer's rule for solving a linear system m.x=b:

For numerical systems, LinearSolve is much faster and more accurate:

Modular computation of a determinant:

Modular determinants:

Recover result:

Shift residue to be symmetric:

Properties & Relations  (5)

CharacteristicPolynomial[m] is equal to :

The determinant is the product of the eigenvalues:

The determinant of a triangular matrix is the product of its diagonal elements:

The determinant of a matrix product is the product of the determinants:

The determinant of the inverse is the reciprocal of the determinant:

Neat Examples  (1)

Determinants of tridiagonal matrices:

Introduced in 1988
 (1.0)