例 2.23

设 $\mathbf{A}$ 为 $n$ 阶矩阵且 ${\mathbf{A}}^m={\mathbf{I}}_n$ ,求证: ${\left({\mathbf{A}}^{\ast }\right)}^m={\mathbf{I}}_n$ .

证明 由 ${\mathbf{A}}^m={\mathbf{I}}_n$ 得 ${\left|\mathbf{A}\right|}^m=1$ . 又 ${\mathbf{A}}^{\ast }=\left|\mathbf{A}\right|{\mathbf{A}}^{-1}$ ,故

$${\left({\mathbf{A}}^{\ast }\right)}^m={\left|\mathbf{A}\right|}^m{\left({\mathbf{A}}^{-1}\right)}^m={\left|\mathbf{A}\right|}^m{\mathbf{A}}^{-m}={\mathbf{I}}_n.$$