# 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, is much faster and more accurate:

Modular computation of a determinant:

Modular determinants:

Recover result:

Shift residue to be symmetric:

## Properties & Relations(5)

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)