考研线性代数：行列式部分初级专项练习题（2024 年）

题目 08

$$
\left|\begin{array}{llll}
3 & 1 & 1 & 1 \\
1 & 3 & 1 & 1 \\
1 & 1 & 3 & 1 \\
1 & 1 & 1 & 3
\end{array}\right|=?
$$

解析 08

$$
\left|\begin{array}{llll}
3 & 1 & 1 & 1 \\
1 & 3 & 1 & 1 \\
1 & 1 & 3 & 1 \\
1 & 1 & 1 & 3
\end{array}\right|=
$$

$$
\left|\begin{array}{llll}
6 & 6 & 6 & 6 \\
1 & 3 & 1 & 1 \\
1 & 1 & 3 & 1 \\
1 & 1 & 1 & 3
\end{array}\right|=
$$

$$
6\left|\begin{array}{llll}
1 & 1 & 1 & 1 \\
1 & 3 & 1 & 1 \\
1 & 1 & 3 & 1 \\
1 & 1 & 1 & 3
\end{array}\right|=
$$

$$
6\left|\begin{array}{llll}
1 & 1 & 1 & 1 \\
0 & 2 & 0 & 0 \\
0 & 0 & 2 & 0 \\
0 & 0 & 0 & 2
\end{array}\right|=
$$

$$
6\left|\begin{array}{lll}
2 & 0 & 0 \\
0 & 2 & 0 \\
0 & 0 & 2
\end{array}\right|=
$$

$$
6 \times 2 \times 2 \times 2=48.
$$