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}+...+B_{0}
where B_{0}= 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]] ] ]

[ [[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)=x^{3}−6*x^{2}+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 :

[[x^{2}−2x+1,x−2,−2x+3],[x−2,x^{2}−4x+4,−x+2],[2x−3,x−2,x^{2}−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)=x^{2}−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]] |