Template:Cauchy–Binet

[math] A \begin{pmatrix} \alpha_{1} & \alpha_{2} & \cdots & \alpha_{k} \\ \beta_{1} & \beta_{2} & \cdots & \beta_{k} \\ \end{pmatrix} = \sum_{\begin{smallmatrix} \gamma_{1} \lt \gamma_{2} \lt \cdots \lt \gamma_{k} \end{smallmatrix}} L \begin{pmatrix} \alpha_{1} & \alpha_{2} & \cdots & \alpha_{k} \\ \gamma_{1} & \gamma_{2} & \cdots & \gamma_{k} \\ \end{pmatrix} U\begin{pmatrix} \gamma_{1} & \gamma_{2} & \cdots & \gamma_{k} \\ \beta_{1} & \beta_{2} & \cdots & \beta_{k} \\ \end{pmatrix} [/math]