Evaluation of Determinants using Minors and Cofactors

by Stephen R. Schmitt

One numerical measure of a matrix called its determinant. The determinant is one of the most important quantities in matrix algebra. When the determinant of a matrix equals zero, the matrix is singular and has no inverse. The determinant can be used as a test for singularity: this is its most important use.

For a 2 by 2 matrix, the determinant is:

The determinant of an n by n matrix A_nn is written as follows:

In order to define the determinant of a matrix of order 3 by 3 or higher, we need to define some additional quantities.

minors

The minor M_ij of the matrix A is the n-1 by n-1 matrix formed by removing the i^th row and the j^th column from matrix A. For example, the minor M₁₂ for the matrix A is:

cofactor

The cofactor C_ij is defined as:

The determinant |A| of an n by n matrix is the calculated as the sum, along any one row or column of A, of array elements multiplied by their corresponding cofactors:

Selecting any row or column will give the same result. Each minor is also a determinant of a square matrix. Each of these can be evaluated recursively using the same procedure. The recursion stops when the dimension reaches 2 x 2 where the simple equation shown above can be used to evaluate the determinant.

Zeno source code for calculating a determinant

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type matrix : array[N,N] of real
    .
    .

%
%  Calculate determinate using recursive 
%  expansion by minors.
%
function det(var a : matrix, n : int) : real

    var i, j, j1, j2 : int
    var d : real := 0
    var m : matrix

    assert (n > 1)

    if n = 2 then
        d := a[1,1]*a[2,2] - a[2,1]*a[1,2]
    else 
        d := 0
        for j1 := 1...n do
            % create minor
            for i := 2...n do
                j2 := 1
                for j := 1...n do
                    continue when j = j1
                    m[i-1,j2] := a[i,j]
                    incr j2
                end for
            end for
            % calculate determinant
            d := d + (-1.0)^(1 + j1)*a[1,j1]*det(m, n-1)
        end for
    end if
    
    return d

end function

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