Maths - Matrix algebra - Determinants 4D

Prerequisites

Determinants are explained here:

Determinants

Description

To find the determinant of:

m00	m01	m02	m03
m10	m11	m12	m13
m20	m21	m22	m23
m30	m31	m32	m33

note: I've started the index numbering from 0 instead of 1 so that the numering is more compatible with array indexing in programming languages.

The Laplace Expansion gives:

m00*

m11	m12	m13
m21	m22	m23
m31	m32	m33

-m01*

m10	m12	m13
m20	m22	m23
m30	m32	m33

+m02*

m10	m11	m13
m20	m21	m23
m30	m31	m33

-m03*

m10	m11	m12
m20	m21	m22
m30	m31	m32

These 3D minor determinants can then be expanded as in the 3D case.

So the value of the determinant for a 4x4 matrix is:

det = m03 * m12 * m21 * m30 - m02 * m13 * m21 * m30-
m03 * m11 * m22 * m30+m01 * m13 * m22 * m30+
m02 * m11 * m23 * m30-m01 * m12 * m23 * m30-
m03 * m12 * m20 * m31+m02 * m13 * m20 * m31+
m03 * m10 * m22 * m31-m00 * m13 * m22 * m31-
m02 * m10 * m23 * m31+m00 * m12 * m23 * m31+
m03 * m11 * m20 * m32-m01 * m13 * m20 * m32-
m03 * m10 * m21 * m32+m00 * m13 * m21 * m32+
m01 * m10 * m23 * m32-m00 * m11 * m23 * m32-
m02 * m11 * m20 * m33+m01 * m12 * m20 * m33+
m02 * m10 * m21 * m33-m00 * m12 * m21 * m33-
m01 * m10 * m22 * m33+m00 * m11 * m22 * m33

The following calculator allows you to calculate the determinant for a 3x3 matrix. Enter the values into the matrix and then press "calc det ->" to display the result:

Code

   // assumes matrix indices start from 0 (0,1,2 and 3)
   public double determinant() {
      double value;
      value =
      m03 * m12 * m21 * m30-m02 * m13 * m21 * m30-m03 * m11 * m22 * m30+m01 * m13 * m22 * m30+
      m02 * m11 * m23 * m30-m01 * m12 * m23 * m30-m03 * m12 * m20 * m31+m02 * m13 * m20 * m31+
      m03 * m10 * m22 * m31-m00 * m13 * m22 * m31-m02 * m10 * m23 * m31+m00 * m12 * m23 * m31+
      m03 * m11 * m20 * m32-m01 * m13 * m20 * m32-m03 * m10 * m21 * m32+m00 * m13 * m21 * m32+
      m01 * m10 * m23 * m32-m00 * m11 * m23 * m32-m02 * m11 * m20 * m33+m01 * m12 * m20 * m33+
      m02 * m10 * m21 * m33-m00 * m12 * m21 * m33-m01 * m10 * m22 * m33+m00 * m11 * m22 * m33;
   return value;
   }

Further Information

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