Formule de Taylor pour le déterminant et Deux applications

AbstractAn explicit formula for the higher differentials of the determinantal function is given. The expression takes a rather elegant form once the exterior powers over the vector space are used, and permits a straightforward determination of the singular locus of the set of matrices of rank ⩽p. Two other applications generalize to arbitrary order matrices the following results: (a) The null matrix is the only matrix A in M(3,K) such that ∀X,Y det(A+X+Y)=det(A+X)+det(A+Y)+det(X+Y)− det A − detX − det Y. (b) If four singular matrices A, B, C, D in M(3,K) satisfy ABCD = 0, then the matrix ABC +ABD + ACD +BCD is also singular