Maths - Matrix Algebra - Determinants

A determinant is a scalar number which is calculated from a matrix. This number can determine whether a set of linear equations are solvable, in other words whether the matrix can be inverted.

Calculating Determinant

The formula for the determinant is shown here:

Notation

This scalar number is represented by the matrix with vertical lines on each side: |M|

Alternative Approaches

Like many mathematical concepts there are are different ways to understand determinants:

Solving linear equations

If we have n equations with n unknowns then we can solve these equations, provided that these equations are all independent, if they are not, then one equation is derived from another and therefore does not provided any additional information.

Geometrical Interpretation

Determinants are perhaps most comonly associated with matricies but it does have a geometric interpretaion that is completely independant of matricies (this geometric interpretation is discussed on this page).

Independence of vectors (orthogonality)

Perhaps we got a clue to this geometric interpretation when we looked at rotations, pure rotations always have a determinant of one. This is related to the situation where we have a set of unit vectors which are mutually othogonal, the determinant of the matrix formed from these vectors is one.

Volume enclosed by vectors

|M| is the volume enclosed by the vectors. However the sign is significant , it may be negative, if an odd number of coordinates are inverted the 'volume' will be negative.

This is related to the tri-vector of a 3D clifford algebra.

Use to calculate inverse matrix

The formula for calculating the inverse of matrix [M] involve multiplication by the scalar factor 1/|M| so if |M| =0 all the components of the inverse will be infinity indicating, in that case, that [M] does not have an inverse.

Evaluation of Determinants using Recursion

We can calculate a n×n determinant from a (n-1)×(n-1) matrix and so on until the determinant of a 1×1 matrix which is just the term itself. This recursive method is also known as expansion by minors.

First some terminology, if we remove one row and one column the remaining determinant is known as a minor and the element at the intersection of the rows and columns being removed is known as the cofactor.

m11 m12 m13
m21 m22 m23
m31 m32 m33

where:

To evaluate the determinant recursively we first choose any row or column, we then make each term in the row or column into a cofactor.

The determinant is calculated from the sum of minors multiplied by their associated cofactors (alternating the signs) taken over the chosen row or column.

|A| = ∑ aij·Cij

where:

Example of a 3x3 Matrix

The determinant of:

m11 m12 m13
m21 m22 m23
m31 m32 m33

is calculated from

  first term second term third term
sign + - +
cofactor m11 m12 m13
minor (remove terms with yellow background)
m11 m12 m13
m21 m22 m23
m31 m32 m33
m11 m12 m13
m21 m22 m23
m31 m32 m33
m11 m12 m13
m21 m22 m23
m31 m32 m33

so we get

m11 m12 m13
m21 m22 m23
m31 m32 m33
= m11
m22 m23
m32 m33
- m12
m21 m23
m31 m33
+ m13
m21 m22
m31 m32

and recursing down to the next level gives:

|M| = m11 m22 m33 + m12 m23 m31 + m13 m21 m32 - m11 m23 m32 - m12 m 21 m33 - m13 m22 m31

Laplace Expansion

We can either expand along any column (j = 1,2 .. n):

  n  
det(M) = Σ Mij Cij
  i=1  

or any row (j = 1,2 .. n):

  n  
det(M) = Σ Mji Cji
  i=1  

where:

Properties of Determinants

All rotation matrices have determinants of 1

For example |R| = cos(a)2 + sin(a)2 = 1

Using determinants to solve simultaneous equations

For 3 unknowns

x
-y
z
-1
m01 m02 v0
m11 m12 v1
m21 m22 v2
m00 m02 v0
m10 m12 v1
m20 m22 v2
m00 m01 v0
m10 m11 v1
m20 m21 v2
m00 m01 m02
m10 m11 m12
m20 m21 m22

Determinants and Clifford Algebra

Ax Ay
Bx By
= A /\ B = pseudoscalar multiplier

Determinants and Clifford algebra are discussed on this page.

Calculating Determinant

The formula for the determinant is shown here:

Code

For Java and C++ code to implement this click here.

Very approximate pseudo code for recursive method:

determinant( A, n )
  if n = 2 then return a[1][1]a[2][2] - a[1][2]a[2][1]
  d := 0 
  for i := 1 to n {
    Minor[x][y] = A[x][y] where x != 1 & y != i 
    d := d + (-1)^(i-1)*a1i*determinant(Minor[1][i],n-1)
  }
  return d

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Terminology and Notation

Specific to this page here:

Determinant represented by the matrix with vertical lines on each side: |M|

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