题目 10
$$
D=\left|\begin{array}{lll}
b+c & c+a & a+b \\
b_{1}+c_{1} & c_{1}+a_{1} & a_{1}+b_{1} \\
b_{2}+c_{2} & c_{2}+a_{2} & a_{2}+b_{2}
\end{array}\right|=?
$$
解析 10
$$
D=\left|\begin{array}{lll}
b+c & c+a & a+b \\
b_{1}+c_{1} & c_{1}+a_{1} & a_{1}+b_{1} \\
b_{2}+c_{2} & c_{2}+a_{2} & a_{2}+b_{2}
\end{array}\right|=
$$
$$
\left|\begin{array}{lll}
b & c+a & a+b \\
b_{1} & c_{1}+a_{1} & a_{1}+b_{1} \\
b_{2} & c_{2}+a_{2} & a_{2}+b_{2}
\end{array}\right|+\left|\begin{array}{ccc}
c & c+a & a+b \\
c_{1} & c_{1}+a_{1} & a_{1}+b_{1} \\
c_{2} & c_{2}+a_{2} & a_{2}+b_{2}
\end{array}\right| =
$$
$$
\left|\begin{array}{ccc}
b & c+a & a \\
b_{1} & c_{1}+a_{1} & a_{1} \\
b_{2} & c_{2}+a_{2} & a_{2}
\end{array}\right|+\left|\begin{array}{ccc}
c & a & a+b \\
c_{1} & a_{1} & a_{1}+b_{1} \\
c_{2} & a_{2} & a_{2}+b_{2}
\end{array}\right|=
$$
$$
\left|\begin{array}{lll}
b & c & a \\
b_{1} & c_{1} & a_{1} \\
b_{2} & c_{2} & a_{2}
\end{array}\right|+\left|\begin{array}{ccc}
c & a & b \\
c_{1} & a_{1} & b_{1} \\
c_{2} & a_{2} & b_{2}
\end{array}\right|=
$$
$$
\left|\begin{array}{lll}
a & b & c \\
a_{1} & b_{1} & c_{1} \\
a_{2} & b_{2} & c_{2}
\end{array}\right|+\left|\begin{array}{lll}
a & b & c \\
a_{1} & b_{1} & c_{1} \\
a_{2} & b_{2} & c_{2}
\end{array}\right|=
$$
$$
2\left|\begin{array}{lll}
a & b & c \\
a_{1} & b_{1} & c_{1} \\
a_{2} & b_{2} & c_{2}
\end{array}\right|.
$$