Adjoint matrix

### 15.2.10 Adjoint matrix

The comatrix of a square matrix A of size n is the matrix B defined by AB=det(A)I. The adjoint matrix Q(x) of A is the comatrix of xI−A. It is a polynomial of degree n−1 in x having matrix coefficients and satisfies:

(xI−A)Q(x)=det(xI−A)I=P(x)I,

where P(x) is the characteristic polynomial of A. Since the polynomial P(x)I−P(A) (with matrix coefficients) is also divisible by x I−A (by algebraic identities), this means that P(A)=0. We also have Q(x)=I xn−1+⋯+B0 where B0 is the comatrix of A (times −1 if n is odd).

The adjoint_matrix command finds the characteristic polynomial and adjoint of a given matrix.

- adjoint_matrix takes A, a square matrix.
- adjoint_matrix(A) returns the list of the coefficients of P(x) (the characteristic polynomial of A), and the list of the matrix coefficients of Q(x) (the adjoint matrix of A).

#### Examples

Let A=[

4 | 1 | −2
1 | 2 | −1
2 | 1 | 0

]. Input:

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

| ⎡ ⎢ ⎢ ⎣ | ⎡ ⎣ | 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)=x3−6x2+12x−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:

⎡ ⎢ ⎢ ⎣ |
x2−2 x+1 | x−2 | −2 x+3
x−2 | x2−4 x+4 | −x+2
2 x−3 | x−2 | x2−6 x+7
| ⎤ ⎥ ⎥ ⎦
| .

Let A=[

4 | 1
1 | 2

]. Input:

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

| ⎡ ⎢ ⎣ | ⎡ ⎣ | 1,−6,7 | ⎤ ⎦ | , | ⎡ ⎢ ⎣ |
⎡ ⎢ ⎣ |
1 | 0
0 | 1
| ⎤ ⎥ ⎦
| , |
⎡ ⎢ ⎣ |
−2 | 1
1 | −4
| ⎤ ⎥ ⎦
| ⎤ ⎥ ⎦ | ⎤ ⎥ ⎦

Hence the characteristic polynomial P is:

P(x)=x2−6x+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
| ⎤ ⎥ ⎦