** suivant:** Companion matrix of a
** monter:** Matrix reduction
** précédent:** Minimal polynomial : pmin
** Table des matières**
** Index**

##

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} + ... + *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]] **]**
]

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} -2*x* + 1, *x* - 2, -2*x* + 3],[*x* - 2, *x*^{2} -4*x* + 4, - *x* + 2],[2*x* - 3, *x* - 2, *x*^{2} - 6*x* + 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]]

** suivant:** Companion matrix of a
** monter:** Matrix reduction
** précédent:** Minimal polynomial : pmin
** Table des matières**
** Index**
giac documentation written by Renée De Graeve and Bernard Parisse