matadjoint

Function: matadjoint
Section: linear_algebra
C-Name: matadjoint0
Prototype: GD0,L,
Help: matadjoint(M,{flag=0}): adjoint matrix of M using Leverrier-Faddeev's
 algorithm. If flag is 1, compute the characteristic polynomial independently
 first.
Doc:
 \idx{adjoint matrix} of $M$, i.e.~a matrix $N$
 of cofactors of $M$, satisfying $M*N=\det(M)*\Id$. $M$ must be a
 (not necessarily invertible) square matrix of dimension $n$.
 If $\fl$ is 0 or omitted, we try to use Leverrier-Faddeev's algorithm,
 which assumes that $n!$ invertible. If it fails or $\fl = 1$,
 compute $T = \kbd{charpoly}(M)$ independently first and return
 $(-1)^{n-1} (T(x)-T(0))/x$ evaluated at $M$.
 \bprog
 ? a = [1,2,3;3,4,5;6,7,8] * Mod(1,4);
 ? matadjoint(a)
 %2 =
 [Mod(1, 4) Mod(1, 4) Mod(2, 4)]

 [Mod(2, 4) Mod(2, 4) Mod(0, 4)]

 [Mod(1, 4) Mod(1, 4) Mod(2, 4)]
 @eprog\noindent
 Both algorithms use $O(n^4)$ operations in the base ring. Over a field,
 they are usually slower than computing the characteristic polynomial or
 the inverse of $M$ directly.
Variant: Also available are
 \fun{GEN}{adj}{GEN x} (\fl=0) and
 \fun{GEN}{adjsafe}{GEN x} (\fl=1).