Formule de Taylor pour le déterminant et deux applications. (Taylor formula for the determinant and two applications) (Q1112912)

Using exterior powers the author gives explicit formulae for higher differentials of the function Det: M(n, K)\(\to K\). The method is based on a generalization to exterior powers of a matrix the formula \(A\cdot^ cA=\det A\cdot I\) where \({}^ cA\) is the adjoint matrix of A. To mention one interesting application: If \(X_ 1\),..., \(X_{n+1}\in M(n\), K) are singular matrices with \(X_ 1\)... \(X_{n+1}=0\) then \(\det (\sum_{i}X_ 1,\quad \hat X_ i,\quad X_{n+1})=0;\) here \(\wedge\) means that the corresponding factor should be omitted.