How to get the cofactor matrix of any given square matrix (singular or non singular)?

Even if the order of matrix is 3x3,4x4 or higher and especially when determinant=0 and we can't use inv().

Answer 1

[John D'Errico]

The matrix of cofactors is moderately well posed, even when the matrix is itself singular. Of course, what you would do with it is your choice. But the simple answer is to just use loops, since each element of the matrix of cofactors is itself just a lower order determaint. But why, when we already have adjoint? For example, the 4x4 magic square produced by magic:

A = magic(4)

A = 4×4

16 2 3 13 5 11 10 8 9 7 6 12 4 14 15 1

rank(A)

ans = 3

While it is a magic square, the matrix is itself singular. So we coiuld not use the determinant of the matrix itself. However, you can just use the adjoint function.

help adjoint

 ADJOINT   Adjoint of the square matrix A.
    X = ADJOINT(A) computes a matrix X of the same dimensions
    as A such that A*X = det(A)*eye(n) = X*A, where n is the number
    of rows of A.
 
    Example:
       X = adjoint(hilb(5))
 
    See also SYM/ADJOINT, DET, INV, RANK.

    Documentation for adjoint
       doc adjoint

    Other functions named adjoint

       sym/adjoint

format short g
adjoint(A)

ans = 4×4

1.0e+00 * -136 -408 408 136 -408 -1224 1224 408 408 1224 -1224 -408 136 408 -408 -136

If A were in fact non-singular, for example:

B = hilb(3)

B = 3×3

1 0.5 0.33333 0.5 0.33333 0.25 0.33333 0.25 0.2

rank(B)

ans = 
     3

adjoint(B)

ans = 3×3

0.0041667 -0.016667 0.013889 -0.016667 0.088889 -0.083333 0.013889 -0.083333 0.083333

We can see that for the non-singular matrix B, the adjoint divided by the determinant does produce an "inverse".

inv(B) - adjoint(B)/det(B)

ans = 3×3

1.0e+00 * 1.954e-14 -7.1054e-14 4.9738e-14 -8.5265e-14 3.6948e-13 -2.8422e-13 7.4607e-14 -3.1264e-13 1.9895e-13

Would you want to use this to compute the inverse? Of course not. You never want to use the determinant to do virtually anything.

[Dev]

Thank you!