Adjoint matrix: adjoint

6.48.10 Adjoint matrix: adjoint_matrix

The comatrix of a square matrix A of size n is the matrix B defined by A× B=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 one argument:
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.

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	−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

⎤
⎥
⎥
⎦

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

⎤
⎥
⎦