方阵积伴随矩阵的一个等式的证明及应用

An equality's proof and application of its adjoint matrix

首先利用矩阵的初等变换给出了伴随矩阵的几个引理,并利用这些引理及初等方阵的理论,对n阶方阵A,B,证明了(AB)*=B*A*,即有关方阵乘积的伴随阵的等式,其证明方法对于工科大学生来说较易接受.此外,应用这一等式,十分简洁地证明了关于伴随矩阵的若干性质.尤其是关于幂等和幂零阵的伴随阵的性质证明.

This paper presents a new method of proving the equality of （AB）^ ＊=B^＊ A^＊ for any square matrix A and B is presented by means of the elementary transformation of matrix and the basic theory of elementary square matrix, which is easier to be accepted by engineering students. By means of the equality it is simpler to prove some properties of the adjoint matrix, especially for the idempotent and nilpotent matrices.