关于微小摄动法的专题讨论

$\bf命题：$$\bf(12浙大一)$设$E$为$n$阶单位阵，$M = \left( {\begin{array}{*{20}{c}}0&E \\ { - E}&0 \end{array}} \right),{A^T}MA = M$，证明：$\left| A \right| = 1$

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$\bf命题：$设$A = {A_{n \times n}},\alpha ,\beta  \in {F^n}$，证明：$\left| {\begin{array}{*{20}{c}}A&\beta \\ {{\alpha ^T}}&k \end{array}} \right| = \left| A \right|k - {\alpha ^T}{A^*}\beta $

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$\bf命题：$