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:

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=[

]. Input:

Hence the characteristic polynomial is:

The determinant of A is equal to −P(0)=8. The comatrix of A is equal to:

Hence the inverse of A is equal to:

The adjoint matrix of A is:

Let A=[

]. Input:

Hence the characteristic polynomial P is:

The determinant of A is equal to P(0)=7. The comatrix of A is equal to

Hence the inverse of A is equal to:

The adjoint matrix of A is: