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Last edited by 青青子衿 at 2019-12-15 14:13:00\begin{gather*}
\det\begin{pmatrix}
a_{\overset{\,}{0}}&a_{\overset{\,}{1}}&a_{\overset{\,}{2}}&\\
&a_{\overset{\,}{0}}&a_{\overset{\,}{1}}&a_{\overset{\,}{2}}\\
b_{\overset{\,}{0}}&b_{\overset{\,}{1}}&b_{\overset{\,}{2}}&\\
&b_{\overset{\,}{0}}&b_{\overset{\,}{1}}&b_{\overset{\,}{2}}\\
\end{pmatrix}\\
\\
=\det\left(
\begin{bmatrix}
b_{\overset{\,}{1}}&b_{\overset{\,}{2}}\\
-b_{\overset{\,}{2}}& 0\\
\end{bmatrix}
\begin{bmatrix}
a_{\overset{\,}{0}}&a_{\overset{\,}{1}}\\
0 &a_{\overset{\,}{0}}\\
\end{bmatrix}-
\begin{bmatrix}
a_{\overset{\,}{1}}&a_{\overset{\,}{2}}\\
-a_{\overset{\,}{2}}&0\\
\end{bmatrix}
\begin{bmatrix}
b_{\overset{\,}{0}}&b_{\overset{\,}{1}}\\
0 &b_{\overset{\,}{0}}\\
\end{bmatrix}\right)\\
\\
=
\,\det\begin{pmatrix}
a_{\overset{\,}{1}}b_{\overset{\,}{0}}-a_{\overset{\,}{0}}b_{\overset{\,}{1}}&a_{\overset{\,}{2}}b_{\overset{\,}{0}}-a_{\overset{\,}{0}}b_{\overset{\,}{2}}\\
a_{\overset{\,}{0}}b_{\overset{\,}{2}}-a_{\overset{\,}{2}}b_{\overset{\,}{0}}&a_{\overset{\,}{1}}b_{\overset{\,}{2}}-\,a_{\overset{\,}{2}}b_{\overset{\,}{1}}\\
\end{pmatrix}
\end{gather*}
\begin{gather*}
\det\begin{pmatrix}
a_{\overset{\,}{0}}&a_{\overset{\,}{1}}&a_{\overset{\,}{2}}&a_{\overset{\,}{3}}&\\
&a_{\overset{\,}{0}}&a_{\overset{\,}{1}}&a_{\overset{\,}{2}}&a_{\overset{\,}{3}}\\
b_{\overset{\,}{0}}&b_{\overset{\,}{1}}&b_{\overset{\,}{2}}&&\\
&b_{\overset{\,}{0}}&b_{\overset{\,}{1}}&b_{\overset{\,}{2}}&\\
&&b_{\overset{\,}{0}}&b_{\overset{\,}{1}}&b_{\overset{\,}{2}}\\
\end{pmatrix}\\
\\
=\det\left(
\begin{bmatrix}
b_{\overset{\,}{1}}&b_{\overset{\,}{2}}& 0 \\
b_{\overset{\,}{2}}& 0 & 0 \\
0 & 0 & 0 \\
\end{bmatrix}
\begin{bmatrix}
a_{\overset{\,}{0}}&a_{\overset{\,}{1}}&a_{\overset{\,}{2}}\\
0 &a_{\overset{\,}{0}}&a_{\overset{\,}{1}}\\
0 & 0 &a_{\overset{\,}{0}}\\
\end{bmatrix}-
\begin{bmatrix}
a_{\overset{\,}{1}}&a_{\overset{\,}{2}}&a_{\overset{\,}{3}}\\
a_{\overset{\,}{2}}&a_{\overset{\,}{3}}& 0 \\
1 & 0 & 0 \\
\end{bmatrix}
\begin{bmatrix}
b_{\overset{\,}{0}}&b_{\overset{\,}{1}}&b_{\overset{\,}{2}}\\
0 &b_{\overset{\,}{0}}&b_{\overset{\,}{1}}\\
0 & 0 &b_{\overset{\,}{0}}\\
\end{bmatrix}\right)\\
\\
=
\,\det\begin{pmatrix}
a_{\overset{\,}{0}}b_{\overset{\,}{1}}-a_{\overset{\,}{1}}b_{\overset{\,}{0}}&a_{\overset{\,}{0}}b_{\overset{\,}{2}}-a_{\overset{\,}{2}}b_{\overset{\,}{0}}&-a_{\overset{\,}{3}}b_{\overset{\,}{0}}\\
a_{\overset{\,}{0}}b_{\overset{\,}{2}}-a_{\overset{\,}{2}}b_{\overset{\,}{0}}&a_{\overset{\,}{1}}b_{\overset{\,}{2}}-\,a_{\overset{\,}{2}}b_{\overset{\,}{1}}-a_{\overset{\,}{3}}b_{\overset{\,}{0}}\,&-a_{\overset{\,}{3}}b_{\overset{\,}{1}}\\
-b_{\overset{\,}{0}}&-b_{\overset{\,}{1}}&-b_{\overset{\,}{2}}\\
\end{pmatrix}
\end{gather*} |
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