两个公式

 

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第一个公式：

这个公式的证明需要用到

第二个公式：

 

证明如下：

\[\begin{array}{l}
\left| {{I_{p \times p}} + {C_{p \times n}}A_{n \times n}^{ - 1}{B_{n \times p}}} \right| = \left| {\begin{array}{*{20}{c}}
{{I_{p \times p}} + {C_{p \times n}}A_{n \times n}^{ - 1}{B_{n \times p}}}&0\\
0&{{I_n}}
\end{array}} \right|\\
\mathop {{\rm{ = = = = = = = = = = }}}\limits^{youcheng\left| {\begin{array}{*{20}{c}}
{{I_{p \times p}}}&0\\
{{B_{n \times p}}}&{{I_n}}
\end{array}} \right|} \left| {\begin{array}{*{20}{c}}
{{I_{p \times p}} + {C_{p \times n}}A_{n \times n}^{ - 1}{B_{n \times p}}}&0\\
{{B_{n \times p}}}&{{I_n}}
\end{array}} \right|\\
\mathop {{\rm{ = = = = = = = = = = = }}}\limits^{zuocheng\left| {\begin{array}{*{20}{c}}
{{I_{p \times p}}}&{ - {C_{p \times n}}A_{n \times n}^{ - 1}}\\
0&{{I_n}}
\end{array}} \right|} \left| {\begin{array}{*{20}{c}}
{{I_{p \times p}} + {C_{p \times n}}A_{n \times n}^{ - 1}{B_{n \times p}} - {C_{p \times n}}A_{n \times n}^{ - 1}{B_{n \times p}}}&{ - {C_{p \times n}}A_{n \times n}^{ - 1}}\\
{{B_{n \times p}}}&{{I_n}}
\end{array}} \right|{\rm{ = }}\left| {\begin{array}{*{20}{c}}
{{I_{p \times p}}}&{ - {C_{p \times n}}A_{n \times n}^{ - 1}}\\
{{B_{n \times p}}}&{{I_n}}
\end{array}} \right|\\
\mathop {{\rm{ = = = = = = = = = = = }}}\limits^{youcheng\left| {\begin{array}{*{20}{c}}
{{I_{p \times p}}}&{{C_{p \times n}}A_{n \times n}^{ - 1}}\\
0&{{I_n}}
\end{array}} \right|} \left| {\begin{array}{*{20}{c}}
{{I_{p \times p}}}&{{C_{p \times n}}A_{n \times n}^{ - 1} - {C_{p \times n}}A_{n \times n}^{ - 1}}\\
{{B_{n \times p}}}&{{I_n}{\rm{ + }}{B_{n \times p}}{C_{p \times n}}A_{n \times n}^{ - 1}}
\end{array}} \right|{\rm{ = }}\left| {\begin{array}{*{20}{c}}
{{I_{p \times p}}}&0\\
{{B_{n \times p}}}&{{I_n}{\rm{ + }}{B_{n \times p}}{C_{p \times n}}A_{n \times n}^{ - 1}}
\end{array}} \right|\\
{\rm{ = }}\left| {\begin{array}{*{20}{c}}
{{I_{p \times p}}}&0\\
{{B_{n \times p}}}&{{I_n}{\rm{ + }}{B_{n \times p}}{C_{p \times n}}}
\end{array}} \right| \cdot \left| {A_{n \times n}^{ - 1}} \right|{\rm{ = }}\left| {A_{n \times n}^{ - 1}} \right| \cdot \left| {{A_{n \times n}} + {B_{n \times p}}{C_{p \times n}}} \right|
\end{array}\]