题目 11
已知 $D=\left|\begin{array}{cccc}
2 & -5 & 1 & 2 \\
-3 & 7 & -1 & 4 \\
5 & -9 & 2 & 7 \\
4 & -6 & 1 & 2
\end{array}\right|$, 则 $D = ?$, $M_{31} + M_{33} + M_{34} = ?$
解析 11
$$
D=\left|\begin{array}{cccc}
2 & -5 & 1 & 2 \\
-3 & 7 & -1 & 4 \\
5 & -9 & 2 & 7 \\
4 & -6 & 1 & 2
\end{array}\right|=
$$
$$
\left|\begin{array}{cccc}
-2 & 1 & 1 & 0 \\
-11 & 19 & -1 & 6 \\
-9 & 12 & 2 & 3 \\
0 & 0 & 1 & 0
\end{array}\right|=
$$
$$
(-1)^{4+3} \cdot\left|\begin{array}{ccc}
-2 & 1 & 0 \\
-11 & 19 & 6 \\
-9 & 12 & 3
\end{array}\right|=
$$
$$
-\left|\begin{array}{ccc}
0 & 1 & 0 \\
27 & 19 & 6 \\
15 & 12 & 3
\end{array}\right| =
$$
$$
\left|\begin{array}{cc}
27 & 6 \\
15 & 3
\end{array}\right|=
$$
$$
27 \times 3-15 \times 6=
$$
$$
(27-30) \times 3=(-3) \times 3=-9.
$$
接着：
$$
M_{31}+M_{33}+M_{34}=
$$
$$
(-1)^{3+1} A_{31}+0 \times A_{32}+(-1)^{3+3} A_{33}+(-1)^{3+4} A_{34}=
$$
$$
1 \times A_{31}+0 \times A_{32}+ 1 \times A_{33}+(-1) \times A_{34}=
$$
$$
\left|\begin{array}{cccc}2 & -5 & 1 & 2 \\ -3 & 7 & -1 & 4 \\ 1 & 0 & 1 & -1 \\ 4 & -6 & 1 & 2\end{array}\right|=\left|\begin{array}{cccc}1 & -5 & 1 & 3 \\ -2 & 7 & -1 & 3 \\ 0 & 0 & 1 & 0 \\ 3 & -6 & 1 & 3\end{array}\right|=
$$
$$
\left|\begin{array}{cccc}1 & -5 & 3 \\ -2 & 7 & 3 \\ 3 & -6 & 3\end{array}\right|=\left|\begin{array}{ccc}1 & -5 & 3 \\ -3 & 12 & 0 \\ 2 & -1 & 0\end{array}\right|=
$$
$$
3 \times \left|\begin{array}{ccc}-3 & 12 \\ 2 & -1\end{array} \right| = 3(3-24)=3 \times(-21)=-63.
$$