Adjoint matrix : adjoint_matrix

adjoint_matrix takes as argument a square matrix A of size n.
adjoint_matrix returns the list of the coefficients of P (the characteristic polynomial of A), and the list of the matrix coefficients of Q (the adjoint matrix of A).

The comatrix of a square matrix A of size n is the matrix B defined by A×B = det(A)×I. The adjoint matrix of A is the comatrix of xI - A. It is a polynomial of degree n - 1 in x having matrix coefficients. The following relation holds:

P(x)×I = det(xI - A)I = (xI - A)Q(x)

Since the polynomial P(x)×I - P(A) (with matrix coefficients) is also divisible by x×I - A (by algebraic identities), this proves that P(A) = 0. We also have Q(x)  =  I×x^n-1 + ... + B₀ where B₀ = is the comatrix of A (up to the sign if n is odd). Input :

adjoint_matrix([[4,1,-2],[1,2,-1],[2,1,0]])

Output :

[ [1,-6,12,-8],
[ [[1,0,0],[0,1,0],[0,0,1]], [[-2,1,-2], [1,-4,-1],[2,1,-6]], [[1,-2,3],[-2,4,2],[-3,-2,7]] ] ]

Hence the characteristic polynomial is :

P(x) = x³ -6*x² + 12*x - 8

The determinant of A is equal to - P(0) = 8. The comatrix of A is equal to :

B = Q(0) = [[1, - 2, 3],[- 2, 4, 2],[- 3, - 2, 7]]

Hence the inverse of A is equal to :

1/8*[[1, - 2, 3],[- 2, 4, 2],[- 3, - 2, 7]]

The adjoint matrix of A is :

[[x² -2x + 1, x - 2, -2x + 3],[x - 2, x² -4x + 4, - x + 2],[2x - 3, x - 2, x² - 6x + 7]]

Input :

adjoint_matrix([[4,1],[1,2]])

Output :

[[1,-6,7],[[[1,0],[0,1]],[[-2,1],[1,-4]]]]

Hence the characteristic polynomial P is :

P(x) = x² - 6*x + 7

The determinant of A is equal to + P(0) = 7. The comatrix of A is equal to

Q(0) = - [[- 2, 1],[1, - 4]]

Hence the inverse of A is equal to :

-1/7*[[- 2, 1],[1, - 4]]

The adjoint matrix of A is :

- [[x - 2, 1],[1, x - 4]]