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.
Examples.
| ⎡ ⎢ ⎢ ⎣ | ⎡ ⎣ | 1,−6,12,−8 | ⎤ ⎦ | , | ⎡ ⎢ ⎢ ⎣ | ⎡ ⎢ ⎢ ⎣ |
| ⎤ ⎥ ⎥ ⎦ | , | ⎡ ⎢ ⎢ ⎣ |
| ⎤ ⎥ ⎥ ⎦ | , | ⎡ ⎢ ⎢ ⎣ |
| ⎤ ⎥ ⎥ ⎦ | ⎤ ⎥ ⎥ ⎦ | ⎤ ⎥ ⎥ ⎦ |
| P(x)=x3−6*x2+12*x−8 |
| B=Q(0)= | ⎡ ⎢ ⎢ ⎣ |
| ⎤ ⎥ ⎥ ⎦ |
| ⎡ ⎢ ⎢ ⎣ |
| ⎤ ⎥ ⎥ ⎦ |
| ⎡ ⎢ ⎢ ⎣ |
| ⎤ ⎥ ⎥ ⎦ |
| ⎡ ⎢ ⎣ | ⎡ ⎣ | 1,−6,7 | ⎤ ⎦ | , | ⎡ ⎢ ⎣ | ⎡ ⎢ ⎣ |
| ⎤ ⎥ ⎦ | , | ⎡ ⎢ ⎣ |
| ⎤ ⎥ ⎦ | ⎤ ⎥ ⎦ | ⎤ ⎥ ⎦ |
| P(x)=x2−6*x+7 |
| Q(0)= − | ⎡ ⎢ ⎣ |
| ⎤ ⎥ ⎦ |
| − |
| ⎡ ⎢ ⎣ |
| ⎤ ⎥ ⎦ |
| − | ⎡ ⎢ ⎣ |
| ⎤ ⎥ ⎦ |