伴随矩阵恒等式的证明

6/8/2020

伴随矩阵最基本的恒等式为: $A(A^*) = |A|E$

怎么证明呢? 参考了这里

直接从定义出发来证明。以一个 3x3 的具体行列式为例。

A = \begin{vmatrix} a_{11} & a_{12} & a_{13} \\\\ a_{21} & a_{22} & a_{23} \\\\ a_{31} & a_{32} & a_{33} \\\\ \end{vmatrix}

根据定义，A 的伴随矩阵 A^*为每个元素的代数余子数按列排组成的矩阵。

A^* = \begin{vmatrix} A_{11} & A_{21} & A_{31} \\\\ A_{12} & A_{22} & A_{32} \\\\ A_{13} & A_{23} & A_{33} \\\\ \end{vmatrix}

所以:

A * A^* = \begin{vmatrix} a_{11} & a_{12} & a_{13} \\\\ a_{21} & a_{22} & a_{23} \\\\ a_{31} & a_{32} & a_{33} \\\\ \end{vmatrix} \begin{vmatrix} A_{11} & A_{21} & A_{31} \\\\ A_{12} & A_{22} & A_{32} \\\\ A_{13} & A_{23} & A_{33} \\\\ \end{vmatrix} = \\\\ \begin{vmatrix} a_{11}A_{11} + a_{12}A_{12} + a_{13}A_{13} & a_{11}A_{21} + a_{12}A_{22} + a_{13}A_{23} & a_{11}A_{31} + a_{12}A_{32} + a_{13}A_{33} \\\\ a_{21}A_{11} + a_{22}A_{12} + a_{23}A_{13} & a_{21}A_{21} + a_{22}A_{22} + a_{23}A_{23} & a_{21}A_{31} + a_{22}A_{32} + a_{23}A_{33} \\\\ a_{31}A_{11} + a_{32}A_{12} + a_{33}A_{13} & a_{31}A_{21} + a_{32}A_{22} + a_{33}A_{23} & a_{31}A_{31} + a_{32}A_{32} + a_{33}A_{33} \\\\ \end{vmatrix} = \\\\ \begin{vmatrix} |A| & 0 & 0 \\\\ 0 & |A| & 0 \\\\ 0 & 0 & |A| \\\\ \end{vmatrix} = |A|E