伴随矩阵的性质：$(\mathit{k}\mathit{A}{)}^{\ast }$（C009）

问题

已知，矩阵 $\mathit{A}$ 是二阶或者大于二阶的方阵，则 $(\mathit{k}\mathit{A}{)}^{\ast }$ $=$ $?$

选项

[A].   $(\mathit{k}\mathit{A}{)}^{\ast }$ $=$ ${\mathit{k}}^{n+1}$ ${\mathit{A}}^{\ast }$

[B].   $(\mathit{k}\mathit{A}{)}^{\ast }$ $=$ ${\mathit{k}}^n$ ${\mathit{A}}^{\ast }$

[C].   $(\mathit{k}\mathit{A}{)}^{\ast }$ $=$ ${\mathit{k}}^{n-1}$ $\mathit{A}$

[D].   $(\mathit{k}\mathit{A}{)}^{\ast }$ $=$ ${\mathit{k}}^{n-1}$ ${\mathit{A}}^{\ast }$

   答 案   

$(\mathit{k}\mathit{A}{)}^{\ast }$ $=$ ${\mathit{k}}^{n-1}$ ${\mathit{A}}^{\ast }$