Adjoint matrix : adjoint_matrix

### 5.50.9 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× xn−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]]