`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 :

Output :

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 :

Output :

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]] |