行列に対する行列式を
や
と表します。
2×2行列、すなわち2行2列
\begin{equation}
A = \left( \begin{array}{cc}
a_{11} & a_{12} \\
a_{21} & a_{22}
\end{array} \right)
\end{equation}の場合、
\begin{eqnarray}
\det A &=& \det \left( \begin{array}{cc}
a_{11} & a_{12} \\
a_{21} & a_{22}
\end{array} \right) \\
|A| &=& \left| \begin{array}{cc}
a_{11} & a_{12} \\
a_{21} & a_{22}
\end{array} \right|
\end{eqnarray}と表し、
\begin{equation}
\det A = |A| = a_{11}a_{22} - a_{12}a_{21}
\end{equation}を意味します。
本稿では、行列の積の行列式が、行列式の積となる、すなわち
\begin{eqnarray}
\det AB &=& \det A \cdot \det B \\
|AB| &=& |A||B|
\end{eqnarray}となることを見ていきます。
なお、
\begin{equation}
B = \left( \begin{array}{cc}
b_{11} & b_{12} \\
b_{21} & b_{22}
\end{array} \right)
\end{equation}とします。
まず、
\begin{eqnarray}
|A| &=& a_{11}a_{22} - a_{12}a_{21} \\
|B| &=& b_{11}b_{22} - b_{12}b_{21}
\end{eqnarray}です。
また、
\begin{eqnarray}
AB &=& \left( \begin{array}{cc}
a_{11} & a_{12} \\
a_{21} & a_{22}
\end{array} \right)
\left( \begin{array}{cc}
b_{11} & b_{12} \\
b_{21} & b_{22}
\end{array} \right) \\
&=& \left( \begin{array}{cc}
a_{11}b_{11}+a_{12}b_{21} & a_{11}b_{12}+a_{12}b_{22} \\
a_{21}b_{11}+a_{22}b_{21} & a_{21}b_{12}+a_{22}b_{22}
\end{array} \right)
\end{eqnarray}なので、
\begin{eqnarray}
|AB| &=& (a_{11}b_{11}+a_{12}b_{21}) (a_{21}b_{12}+a_{22}b_{22}) - (a_{11}b_{12}+a_{12}b_{22}) (a_{21}b_{11}+a_{22}b_{21}) \\
&=& a_{11}a_{21}b_{11}b_{12} + a_{11}a_{22}b_{11}b_{22} + a_{12}a_{21}b_{12}b_{21} + a_{12}a_{22}b_{21}b_{22} \\
&& - a_{11}a_{21}b_{11}b_{12} - a_{11}a_{22}b_{12}b_{21} - a_{12}a_{21}b_{11}b_{22} - a_{12}a_{22}b_{21}b_{22} \\
&=& a_{11}a_{22}b_{11}b_{22} + a_{12}a_{21}b_{12}b_{21} - a_{11}a_{22}b_{12}b_{21} - a_{12}a_{21}b_{11}b_{22} \\
&=& a_{11}a_{22}(b_{11}b_{22} - b_{12}b_{21}) - a_{12}a_{21}(b_{11}b_{22} - b_{12}b_{21}) \\
&=& (a_{11}a_{22} - a_{12}a_{21})(b_{11}b_{22} - b_{12}b_{21}) \\
&=& |A||B|
\end{eqnarray}となります。
強引に計算していくと重複項は相殺され、あとは因数分解をすれば導くことができます。
