# 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:

The determinant of an exact matrix:

## Scope(10)

### Basic Uses(6)

Find the determinant of a MachinePrecision matrix:

Determinant of a complex matrix:

Determinant of an exact matrix:

Determinant of an arbitrary-precision matrix:

Determinant of a symbolic matrix:

The determinant of a large numerical matrix is computed efficiently:

Note that the result may not be a machine number:

### Special Matrices(4)

Determinants of sparse matrices:

Determinants of structured matrices:

IdentityMatrix always has unit determinant:

Determinant of HilbertMatrix:

## Options(1)

### Modulus(1)

Compute a determinant using arithmetic modulo 47:

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

## Applications(19)

### Area and Volumes(6)

Use Det to find area of a parallelogram spanned by and :

Visualize the parallelogram when one vertex is at the origin:

The area is given by the absolute value of the determinant:

Compare with the result given by Area:

Use Det to find volume of a parallelepiped spanned by , and :

Visualize the parallelepiped when one vertex is at the origin:

The volume is given by the absolute value of the determinant:

Compare with a direct computation using Volume:

Use Det to find hypervolume of a hyper-parallelepiped spanned by the following vectors:

The hypervolume is given by the absolute value of the determinant:

Compare with the result given by RegionMeasure:

The determinant itself is negative, so the are not right-handed:

Simply reorder any two vectors, say the middle two, to produce a right-handed set:

Find the area of the image of the unit disk under the linear transformation associated to the matrix :

The area of the image is given by :

Compare with a direct computation:

Visualize the image :

Find the volume factor in the change of variables formula between Cartesian and polar coordinates. The mapping from polar to Cartesian coordinates is given by:

Compute the Jacobian of the mapping using Grad:

By the change of variables theorem, the volume is the determinant of the Jacobian:

Compare with the result given by CoordinateChartData:

The same procedure will work with any coordinate system, for example, spherical coordinates:

Use the change of variables theorem to compute , where is the following region:

First, define hyperbolic coordinates as follows:

The region clearly corresponds to and . By the change of variables formula, . The gradient is given by:

The determinant of the gradient is twice the function whose integral is :

Hence, is given by the trivial integral :

Compare with a direct integration over the region:

### Orientation and Rotations(5)

Determine whether the following basis for is right-handed:

The determinant of the matrix formed by the basis is negative, so it is not right-handed:

Determine if linear transformation corresponding to is orientation-preserving or orientation-reversing:

As , the mapping is orientation-preserving:

Show that the following matrix is not a rotation matrix:

All rotation matrices have unit determinant; since , it cannot be a rotation matrix:

Show that the matrix is orthogonal and determine if it is a rotation matrix or includes a reflection:

Up to the input precision, , which shows that is orthogonal:

All orthogonal matrices have , but rotations have ; as , includes a reflection:

The generalization of a rotation matrix to complex vector spaces is a special unitary matrix that is unitary and has unit determinant. Show that the following matrix is a special unitary matrix:

The matrix is unitary because :

It also has unit determinant, so it is in fact an element of the special unitary group :

### Linear and Abstract Algebra(8)

Determine the values of the parameter for which the system , has a unique solution and describe that solution. First, form the coefficient matrix and constant vector :

The solutions will be unique as :

Solving over the reals gives three open intervals separated at and :

Since the matrix is invertible for these values of , the solution is simply :

Verify the solution in the original system of equations:

Use Cramer's rule to solve the system of equations , , . First, form the coefficient matrix and constant vector :

Form the three matrices where replaces the the corresponding columns of :

The entries of the solution are given by :

Verify the result:

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

Use the function to solve a system for particular values of m and b:

Verify the solution:

For numerical systems, is much faster and more accurate:

Determine if the matrix has a nontrivial kernel (null space):

Since the determinant is nonzero, the kernel is trivial:

Confirm the result using NullSpace:

Determine if the mapping corresponding to the matrix is injective:

Since , the mapping is not injective:

Confirm the result using FunctionInjective:

Since defines a linear function , the failure to be injective implies a failure to be surjective:

Determine if the matrix defines an automorphism (a bijective linear map):

Since , the mapping is an automorphism:

Confirm the result using FunctionBijective:

Compute the cofactor obtained from removing row i and column j:

Check the result:

Modular computation of a determinant:

Modular determinants:

Recover the result:

Shift the residue to be symmetric:

Confirm that the non-modular determinant was recovered:

## Properties & Relations(14)

The determinant is the product of the eigenvalues:

Det satisfies , where is all -permutations and is Signature:

Det can be computed recursively via cofactor expansion along any row:

Or any column:

The determinant is the signed volume of the parallelepiped generated by its rows:

This equals the volume up to sign:

A square matrix has an inverse if and only if its determinant is nonzero: 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:

A matrix and its transpose have equal determinants:

The determinant of the matrix exponential is the exponential of the trace:

is equal to :

Det[m] can be computed from :

Consider two rectangular matrices and such that and are both square:

Sylvester's determinant theorem states that , where is the matching identity matrix:

If a matrix is the TensorProduct of two vectors and , then :

This can be expressed equally in terms of KroneckerProduct:

This follows from Sylvester's determinant theorem for the corresponding row and column matrices:

## Neat Examples(1)

Determinants of tridiagonal matrices:

A closed-form formula for these determinants is given by :