陈现平张彬高等代数考研所有例题参考解答(更新于240505)
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# 行列式
## 行列式的计算方法57题
行列式计算的原则
对于以数字为元素的行列式计算, 可以先观察规律, 若无规律, 一般是先选定某一行 (列), 利用行列式的性质, 将其中的元素尽可能地化为 0, 然后按这一行 (列) 展开, 如此继续下去, 即可得结果.
对于以字母为元素的行列式计算, 一般首先弄清行列式中元素的结构, 找出规律, 然后充分利用行列式的性质, 化为三角形行列式或利用降阶法找出递推公式.
计算行列式有如下口决:认清元素, 分析结构, 先看特殊, 再想一般, 熟用性质, 必要展开. [这口诀记不住啊, 没关系, 多做题目就是...]
### 化三角法
化三角形法就是利用行列式的性质将所求的行列式化为上 (下) 三角形行列式计算. [行列式化出一大片的零出来, 即使不是三角形, 也是准对角的, 降阶了.]
例 1.1.1 (广西民族大学, 2021) 计算 $\displaystyle n$ 阶行列式
$\displaystyle \tiny\boxed{@跟锦数学微信公众号}$
$$ \begin{aligned} D=\left|\begin{array}{cccccc}
1 & 2 & 3 & \cdots & n-1 & n \\
1 & -1 & 0 & \cdots & 0 & 0 \\
0 & 2 & -2 & \cdots & 0 & 0 \\
\vdots & \vdots & \vdots & & \vdots & \vdots \\
0 & 0 & 0 & \cdots & -(n-2) & 0 \\
0 & 0 & 0 & \cdots & n-1 & -(n-1)
\end{array}\right|.\tiny\boxed{\begin{array}{c}\mbox{跟锦数学微信公众号}\\\mbox{www.zhangzujin.cn}\end{array}}\end{aligned} $$
[例 1.1.01 (广西民族大学, 2021)](https://mp.weixin.qq.com/s?__biz=MzU3OTExMzI1Mg==&mid=2247541867&idx=4&sn=57cc99c49866e82498c656b56c48baa0&chksm=fd697c20ca1ef5365ac94bbac985e83b63068db0901e9f7b9647163a7eb8077c78a9c9ce723e)
例 1.1.2 (武汉大学,2020) 计算 $\displaystyle n$ 阶行列式
$\displaystyle \tiny\boxed{@跟锦数学微信公众号}$
$$ \begin{aligned} \left|\begin{array}{cccccc}
0 & 1 & 2 & 3 & \cdots & n-1 \\
1 & 0 & 1 & 2 & \cdots & n-2 \\
2 & 1 & 0 & 1 & \cdots & n-3 \\
3 & 2 & 1 & 0 & \cdots & n-4 \\
\vdots & \vdots & \vdots & \vdots & & \vdots \\
n-1 & n-2 & n-3 & n-4 & \cdots & 0
\end{array}\right|.\tiny\boxed{\begin{array}{c}\mbox{跟锦数学微信公众号}\\\mbox{www.zhangzujin.cn}\end{array}}\end{aligned} $$
[例 1.1.02 (武汉大学,2020)](https://mp.weixin.qq.com/s?__biz=MzU3OTExMzI1Mg==&mid=2247541867&idx=3&sn=773d3cc2b78c7ae49e3df11de256f4c6&chksm=fd697c20ca1ef53692f23403073ba1d4f9d8fc6893447b82afb3879876131924aa6813aee832)
例 1.1.3 (复旦大学高等代数每周一题 [问题 2021A01]; 中国科学院,2015) 求下列 $\displaystyle n$ 阶行列式的值:
$\displaystyle \tiny\boxed{@跟锦数学微信公众号}$
$$ \begin{aligned} |A|=\left|\begin{array}{ccccc}
1 & a & a^2 & \cdots & a^{n-1} \\
a & 1 & a & \cdots & a^{n-2} \\
a^2 & a & 1 & \cdots & a^{n-3} \\
\vdots & \vdots & \vdots & & \vdots \\
a^{n-1} & a^{n-2} & a^{n-3} & \cdots & 1
\end{array}\right|.\tiny\boxed{\begin{array}{c}\mbox{跟锦数学微信公众号}\\\mbox{www.zhangzujin.cn}\end{array}}\end{aligned} $$
[例 1.1.03 (复旦大学高等代数每周一题 [问题 2021A01]; 中国科学院,2015)](https://mp.weixin.qq.com/s?__biz=MzU3OTExMzI1Mg==&mid=2247541867&idx=2&sn=2621546d0b98d75db1008e16356406b4&chksm=fd697c20ca1ef536f8b79db30381a8ebb21eac6dc6a9ba8b9bb3516c778c653c55190afe8ca3)
; [例 1.1.03 (复旦大学高等代数每周一题 [问题 2021A01]; 中国科学院,2015)另解](https://mp.weixin.qq.com/s?__biz=MzU3OTExMzI1Mg==&mid=2247541879&idx=8&sn=ee9a080c2595c2da68cebeed976ee014&chksm=fd697c3cca1ef52a8680433a8d7d952d74cb0842f1a87d4330291091b8b5161e45fff8ce8a96)
例 1.1.4 (中南大学,2023) 计算 $\displaystyle n+1$ 阶行列式
$\displaystyle \tiny\boxed{@跟锦数学微信公众号}$
$$ \begin{aligned} D_{n+1}=\left|\begin{array}{cccccc}
a & -1 & 0 & 0 & \ldots & 0 \\
a x & a & -1 & 0 & \ldots & 0 \\
a x^2 & a x & a & -1 & \ldots & 0 \\
\vdots & \vdots & \vdots & \vdots & & \vdots \\
a x^{n-1} & a x^{n-2} & a x^{n-3} & a x^{n-4} & \ldots & -1 \\
a x^n & a x^{n-1} & a x^{n-2} & a x^{n-3} & \ldots & a
\end{array}\right|.\tiny\boxed{\begin{array}{c}\mbox{跟锦数学微信公众号}\\\mbox{www.zhangzujin.cn}\end{array}}\end{aligned} $$
[例 1.1.04 (中南大学,2023)](https://mp.weixin.qq.com/s?__biz=MzU3OTExMzI1Mg==&mid=2247541867&idx=1&sn=76d1eca3dad5092d43fd4160ef33348c&chksm=fd697c20ca1ef5361be30a9216f64e613ed5a542a45049a0efe5d095b267ce1b123f63d880a3)
### 降阶法
降阶法就是利用行列式的性质、拉普拉斯 (Laplace) 定理、行列式降阶定理降低行列式的阶数,然后计算.
例 1.1.5 (中国石油大学,2021; 北京邮电大学,2021) 计算 $\displaystyle n$ 阶行列式
$\displaystyle \tiny\boxed{@跟锦数学微信公众号}$
$$ \begin{aligned} D_n=\left|\begin{array}{ccccc}
\lambda & a & a & \cdots & a \\
b & \alpha & \beta & \cdots & \beta \\
b & \beta & \alpha & \cdots & \beta \\
\vdots & \vdots & \vdots & & \vdots \\
b & \beta & \beta & \cdots & \alpha
\end{array}\right|.\tiny\boxed{\begin{array}{c}\mbox{跟锦数学微信公众号}\\\mbox{www.zhangzujin.cn}\end{array}}\end{aligned} $$
[例 1.1.05 (中国石油大学,2021; 北京邮电大学,2021)](https://mp.weixin.qq.com/s?__biz=MzU3OTExMzI1Mg==&mid=2247541879&idx=6&sn=d04d2a527670abf20fa27f40a727048d&chksm=fd697c3cca1ef52aa61e38637411cdd58f7cecebcd392e35afeabf46874daee362f2511b9073)
例 1.1.6 (西安建筑科技大学,2018; 南昌大学,2020; 沈阳工业大学,2021) 计算 $\displaystyle n$ 阶行列式
$\displaystyle \tiny\boxed{@跟锦数学微信公众号}$
$$ \begin{aligned} D_n=\left|\begin{array}{cccccc}
x & -1 & 0 & \cdots & 0 & 0 \\
0 & x & -1 & \cdots & 0 & 0 \\
0 & 0 & x & \cdots & 0 & 0 \\
\vdots & \vdots & \vdots & & \vdots & \vdots \\
0 & 0 & 0 & \cdots & x & -1 \\
a_n & a_{n-1} & a_{n-2} & \cdots & a_2 & x+a_1
\end{array}\right|.\tiny\boxed{\begin{array}{c}\mbox{跟锦数学微信公众号}\\\mbox{www.zhangzujin.cn}\end{array}}\end{aligned} $$
[例 1.1.06 (西安建筑科技大学,2018; 南昌大学,2020; 沈阳工业大学,2021)](https://mp.weixin.qq.com/s?__biz=MzU3OTExMzI1Mg==&mid=2247541879&idx=5&sn=e5495cf96b3e3431ca2504032ffab256&chksm=fd697c3cca1ef52ae8037286c8e63ee8bb70eb104db109eda0c348f32e69cb0be31fb0b25297)
例 1.1.7 (重庆大学, 2022) 计算行列式
$\displaystyle \tiny\boxed{@跟锦数学微信公众号}$
$$ \begin{aligned} \left|\begin{array}{crrrrr}
a_1 & -1 & 0 & \cdots & 0 & 0 \\
a_2 & x & -1 & \cdots & 0 & 0 \\
a_3 & 0 & x & \cdots & 0 & 0 \\
\vdots & \vdots & \vdots & & \vdots & \vdots \\
a_{n-1} & 0 & 0 & \cdots & x & -1 \\
a_n & 0 & 0 & \cdots & 0 & x
\end{array}\right|.\tiny\boxed{\begin{array}{c}\mbox{跟锦数学微信公众号}\\\mbox{www.zhangzujin.cn}\end{array}}\end{aligned} $$
[例1.1.07(重庆大学,2022)](https://mp.weixin.qq.com/s?__biz=MzU3OTExMzI1Mg==&mid=2247542031&idx=6&sn=a7baebb7ae9b8018dc6940fc2daced48&chksm=fd697d44ca1ef45218007c194f0ab5579e096c794f600b765890c41205d41ce8c6a91e37db7d)
例 1.1.8 (重庆工学院,2009) 设 $\displaystyle f(x)$ 是一个整系数多项式, 且
$\displaystyle \tiny\boxed{@跟锦数学微信公众号}$
$$ \begin{aligned} f(x)=\left|\begin{array}{rrrrcc}
x & -1 & 0 & \cdots & 0 & 0 \\
0 & x & -1 & \cdots & 0 & 0 \\
0 & 0 & x & \cdots & 0 & 0 \\
\vdots & \vdots & \vdots & & \vdots & \vdots \\
0 & 0 & 0 & \cdots & x & -1 \\
3 & 3^2 & 3^3 & \cdots & 3^{n-1} & 3^n+x
\end{array}\right|,\tiny\boxed{\begin{array}{c}\mbox{跟锦数学微信公众号}\\\mbox{www.zhangzujin.cn}\end{array}}\end{aligned} $$
其中 $\displaystyle n \geqslant 2$ . 证明:$\displaystyle f(x)$ 在有理数域上不可约.
[例 1.1.08 (重庆工学院,2009)](https://mp.weixin.qq.com/s?__biz=MzU3OTExMzI1Mg==&mid=2247541879&idx=3&sn=03a7b31c2fecdddf462c18161870df11&chksm=fd697c3cca1ef52a7a6faa7ebd88e20f950dd0ba41123b7689581328e2a97cbaae8f7fecdda2)
例 1.1.9 (首都师范大学,2021) 求行列式
$\displaystyle \tiny\boxed{@跟锦数学微信公众号}$
$$ \begin{aligned}
D=\left|\begin{array}{cccccccccc}
17&&&&&&&&&18\\
&13&&&&&&&14&\\
&&9&&&&&10&&\\
&&&5&&&6&&&\\
&&&&1&2&&&&\\
&&&&3&4&&&&\\
&&&7&&&8&&&\\
&&11&&&&&12&&\\
&15&&&&&&&16&\\
19&&&&&&&&&20
\end{array}\right|.
\tiny\boxed{\begin{array}{c}\mbox{跟锦数学微信公众号}\\\mbox{www.zhangzujin.cn}\end{array}}\end{aligned} $$
[例 1.1.09 (首都师范大学,2021)](https://mp.weixin.qq.com/s?__biz=MzU3OTExMzI1Mg==&mid=2247541879&idx=2&sn=0d9178a91fd30d6e6260eff3563596f6&chksm=fd697c3cca1ef52ae9ba120c55626b1ee6f54c316466dab522346a08973bde556740174ef3ea)
例 1.1.10 (南京师范大学, 2023) 计算 $\displaystyle 2 n$ 阶行列式
$\displaystyle \tiny\boxed{@跟锦数学微信公众号}$
$$ \begin{aligned} D_{2 n}=\left|\begin{array}{cccccc}
a_n & & & & & b_n \\
& \ddots & & & \mathrm{id}dots & \\
& & a_1 & b_1 & & \\
& & c_1 & d_1 & & \\
&\mathrm{id}dots & & & \ddots & \\
c_n & & & & & d_n
\end{array}\right|.\tiny\boxed{\begin{array}{c}\mbox{跟锦数学微信公众号}\\\mbox{www.zhangzujin.cn}\end{array}}\end{aligned} $$
[例 1.1.10 (首都师范大学,2021)](https://mp.weixin.qq.com/s?__biz=MzU3OTExMzI1Mg==&mid=2247541879&idx=1&sn=460241bc784bd7c2fdd136e41789e366&chksm=fd697c3cca1ef52abb689eae1ecef66fe430753ceaa1aa83259f93f637b9d5160f5ec3cad989)
例 1.1.11 计算 $\displaystyle 2 n+1$ 阶行列式
$\displaystyle \tiny\boxed{@跟锦数学微信公众号}$
$$ \begin{aligned} D_{2 n+1}=\left|\begin{array}{ccccccc}
a_n & & & & & & b_n \\
& \ddots & & & & \mathrm{id}dots & \\
& & a_1 & & b_1 & & \\
& & & e & & & \\
& & c_1 & & d_1 & & \\
& \mathrm{id}dots & & & & \ddots & \\
c_n & & & & & & d_n
\end{array}\right|.\tiny\boxed{\begin{array}{c}\mbox{跟锦数学微信公众号}\\\mbox{www.zhangzujin.cn}\end{array}}\end{aligned} $$
[例 1.1.11](https://mp.weixin.qq.com/s?__biz=MzU3OTExMzI1Mg==&mid=2247541903&idx=7&sn=d4b3e289323bf32b031d2f50d6fc5464&chksm=fd697cc4ca1ef5d216eaea14d56e5303033de8d506e7b722f5ce8779e6ea2e7295e9e1235112)
### 加边法
加边法 (也称为升阶法) 是指在原行列式的基础上增加多行多列, 通常增加一行一列, 但新的行列式更易计算.
例 1.1.12 (杭州电子科技大学,2021) 计算 $\displaystyle n$ 阶行列式
$\displaystyle \tiny\boxed{@跟锦数学微信公众号}$
$$ \begin{aligned} D_n=\left|\begin{array}{cccc}
1+a & 2 & \cdots & n \\
1 & 2+a & \cdots & n \\
\vdots & \vdots & & \vdots \\
1 & 2 & \cdots & n+a
\end{array}\right|.\tiny\boxed{\begin{array}{c}\mbox{跟锦数学微信公众号}\\\mbox{www.zhangzujin.cn}\end{array}}\end{aligned} $$
[例 1.1.12 (杭州电子科技大学,2021)](https://mp.weixin.qq.com/s?__biz=MzU3OTExMzI1Mg==&mid=2247541903&idx=6&sn=c7915969e4d069f9ba8b75cdcbaf78bb&chksm=fd697cc4ca1ef5d2459fc9285c3f055614d779be2c3689e09b874f31dd8226697623b30f56b6)
例 1.1.13 (广东财经大学,2022) 计算行列式
$\displaystyle \tiny\boxed{@跟锦数学微信公众号}$
$$ \begin{aligned} D=\left|\begin{array}{ccccc}
1+x_1 & x_2 & x_3 & \cdots & x_n \\
x_1 & 1+x_2 & x_3 & \cdots & x_n \\
x_1 & x_2 & 1+x_3 & \cdots & x_n \\
\vdots & \vdots & \vdots & & \vdots \\
x_1 & x_2 & x_3 & \cdots & 1+x_n
\end{array}\right|.\tiny\boxed{\begin{array}{c}\mbox{跟锦数学微信公众号}\\\mbox{www.zhangzujin.cn}\end{array}}\end{aligned} $$
[例 1.1.13 (广东财经大学,2022)](https://mp.weixin.qq.com/s?__biz=MzU3OTExMzI1Mg==&mid=2247541903&idx=5&sn=da27cb64484aa563b3ceed5b4e383c84&chksm=fd697cc4ca1ef5d25d1589756792442139fdd8ff7f5f96b4903c3060ffae3057a3cfc61b7bcd)
例 1.1.14 (北京科技大学,2008; 华东理工大学,2021) 求行列式
$\displaystyle \tiny\boxed{@跟锦数学微信公众号}$
$$ \begin{aligned} D_n=\left|\begin{array}{ccccc}
1 & 2 & \cdots & n-1 & n+x \\
1 & 2 & \cdots & n-1+x & n \\
\vdots & \vdots & & \vdots & \vdots \\
1+x & 2 & \cdots & n-1 & n
\end{array}\right|.\tiny\boxed{\begin{array}{c}\mbox{跟锦数学微信公众号}\\\mbox{www.zhangzujin.cn}\end{array}}\end{aligned} $$
[例 1.1.14 (北京科技大学,2008; 华东理工大学,2021)](https://mp.weixin.qq.com/s?__biz=MzU3OTExMzI1Mg==&mid=2247541903&idx=4&sn=a6e26645c76edfac21670b2133fe7a80&chksm=fd697cc4ca1ef5d2d61bf83a226a733c4c7a3f3a8057bf175ea9b25adc02a6b64446578e1f69)
例 1.1.15 计算行列式
$\displaystyle \tiny\boxed{@跟锦数学微信公众号}$
$$ \begin{aligned} D=\left|\begin{array}{cccc}
x_1 & y & \cdots & y \\
y & x_2 & \cdots & y \\
\vdots & \vdots & & \vdots \\
y & y & \cdots & x_n
\end{array}\right|,\tiny\boxed{\begin{array}{c}\mbox{跟锦数学微信公众号}\\\mbox{www.zhangzujin.cn}\end{array}}\end{aligned} $$
其中 $\displaystyle x_i-y \neq 0(i=1,2, \cdots, n)$ .
[例 1.1.15](https://mp.weixin.qq.com/s?__biz=MzU3OTExMzI1Mg==&mid=2247541903&idx=3&sn=d3ea75f1e6be687c72fa095a32ed2954&chksm=fd697cc4ca1ef5d279aba32372af9b2fd5cf10a8020fe9f243dbfeff8971e7a6feca1382d3c0)
例 1.1.16 (东北大学,2021) 计算行列式
$\displaystyle \tiny\boxed{@跟锦数学微信公众号}$
$$ \begin{aligned} D_n=\left|\begin{array}{cccc}
x_1+x & x_2 & \cdots & x_n \\
x_1 & x_2+x & \cdots & x_n \\
\vdots & \vdots & & \vdots \\
x_1 & x_2 & \cdots & x_n+x
\end{array}\right|.\tiny\boxed{\begin{array}{c}\mbox{跟锦数学微信公众号}\\\mbox{www.zhangzujin.cn}\end{array}}\end{aligned} $$
[例 1.1.16 (东北大学,2021)](https://mp.weixin.qq.com/s?__biz=MzU3OTExMzI1Mg==&mid=2247541903&idx=2&sn=599b4ccd3951b215b3f33e8fb3c10a2b&chksm=fd697cc4ca1ef5d2ad4c986c99ad7336ce286d57f552075574e62765503580b417f5334fa375)
例 1.1.17 (天津大学,2021; 兰州大学,2021; 南昌大学,2021) 计算行列式
$\displaystyle \tiny\boxed{@跟锦数学微信公众号}$
$$ \begin{aligned} \left|\begin{array}{ccccc}
a_1+x_1 & a_2 & a_3 & \cdots & a_n \\
a_1 & a_2+x_2 & a_3 & \cdots & a_n \\
a_1 & a_2 & a_3+x_3 & \cdots & a_n \\
\vdots & \vdots & \vdots & & \vdots \\
a_1 & a_2 & a_3 & \cdots & a_n+x_n
\end{array}\right|.\tiny\boxed{\begin{array}{c}\mbox{跟锦数学微信公众号}\\\mbox{www.zhangzujin.cn}\end{array}}\end{aligned} $$
[例 1.1.17 (天津大学,2021; 兰州大学,2021; 南昌大学,2021)](https://mp.weixin.qq.com/s?__biz=MzU3OTExMzI1Mg==&mid=2247541903&idx=1&sn=045098a561df2b196f2017b8df4ef061&chksm=fd697cc4ca1ef5d2501c50ab43beaa2f58b4fb99abcf7f9fb92f3e3c0709f65409d17672910a)
例 1.1.18 (首都师范大学,2015) 求行列式
$\displaystyle \tiny\boxed{@跟锦数学微信公众号}$
$$ \begin{aligned} \left|\begin{array}{cccc}
1 & 1 & 1 & 1 \\
a & b & c & d \\
a^2 & b^2 & c^2 & d^2 \\
a^4 & b^4 & c^4 & d^4
\end{array}\right|.\tiny\boxed{\begin{array}{c}\mbox{跟锦数学微信公众号}\\\mbox{www.zhangzujin.cn}\end{array}}\end{aligned} $$
[例1.1.18(首都师范大学,2015)](https://mp.weixin.qq.com/s?__biz=MzU3OTExMzI1Mg==&mid=2247541955&idx=6&sn=dda65a7db0b62107285a920ad13d0fe0&chksm=fd697c88ca1ef59e210f4d3afcc194939479f8d80c71b6bed513089e49bf8e3b2a4c6c26d907)
例 1.1.19 (曲阜师范大学, 2023) 计算如下行列式:
$\displaystyle \tiny\boxed{@跟锦数学微信公众号}$
$$ \begin{aligned} \left|\begin{array}{ccccc}
1 & 1 & 1 & 1 & 1 \\
1 & 2 & 3 & 4 & 5 \\
1 & 4 & 9 & 16 & 25 \\
1 & 8 & 27 & 64 & 125 \\
1 & 32 & 243 & 1024 & 3125
\end{array}\right|.\tiny\boxed{\begin{array}{c}\mbox{跟锦数学微信公众号}\\\mbox{www.zhangzujin.cn}\end{array}}\end{aligned} $$
[例1.1.19(曲阜师范大学,2023)](https://mp.weixin.qq.com/s?__biz=MzU3OTExMzI1Mg==&mid=2247541955&idx=5&sn=7613dbd56b1d47e82df9ce4a2b18e601&chksm=fd697c88ca1ef59efaf8ce2112c5752f1da3e1eb64e4b6d28bc29e56f593e1d687ccf10bdf89)
例 1.1.20 (汕头大学,2019) 计算行列式
$\displaystyle \tiny\boxed{@跟锦数学微信公众号}$
$$ \begin{aligned} D_n=\left|\begin{array}{ccccc}
1 & 1 & 1 & \cdots & 1 \\
a_1^2 & a_2^2 & a_3^2 & \cdots & a_n^2 \\
a_1^3 & a_2^3 & a_3^3 & \cdots & a_n^3 \\
\vdots & \vdots & \vdots & & \vdots \\
a_1^n & a_2^n & a_3^n & \cdots & a_n^n
\end{array}\right|.\tiny\boxed{\begin{array}{c}\mbox{跟锦数学微信公众号}\\\mbox{www.zhangzujin.cn}\end{array}}\end{aligned} $$
[例1.1.20(汕头大学,2019)](https://mp.weixin.qq.com/s?__biz=MzU3OTExMzI1Mg==&mid=2247541955&idx=4&sn=648af79685fcdd9d906b645f846e9447&chksm=fd697c88ca1ef59e85189f865ad3372bc7d7f4aba28b7edc3885355f0374cb620468785d6a28)
例 1.1.21 (湖南大学,2023) 计算 $\displaystyle n$ 阶行列式
$\displaystyle \tiny\boxed{@跟锦数学微信公众号}$
$$ \begin{aligned} D_n=\left|\begin{array}{cccc}
1 & 1 & \cdots & 1 \\
1 & 2 & \cdots & n \\
1 & 2^2 & \cdots & n^2 \\
\vdots & \vdots & & \vdots \\
1 & 2^{n-2} & \cdots & n^{n-2} \\
1 & 2^n & \cdots & n^n
\end{array}\right|.\tiny\boxed{\begin{array}{c}\mbox{跟锦数学微信公众号}\\\mbox{www.zhangzujin.cn}\end{array}}\end{aligned} $$
[例1.1.21(湖南大学,2023)](https://mp.weixin.qq.com/s?__biz=MzU3OTExMzI1Mg==&mid=2247541955&idx=3&sn=49c67b5f636f3214639e675ff90853cb&chksm=fd697c88ca1ef59e5b5961347eb9a2cefb7024e8544048ec89d5cbc065b14c89581d97723b74)
例 1.1.22 (湘潭大学,2023) 求行列式
$\displaystyle \tiny\boxed{@跟锦数学微信公众号}$
$$ \begin{aligned} D=\left|\begin{array}{ccccccc}
1 & a_1 & a_1^2 & \cdots & a_1^{n-3} & a_1^{n-1} & a_1^n \\
1 & a_2 & a_2^2 & \cdots & a_2^{n-3} & a_2^{n-1} & a_2^n \\
\vdots & \vdots & \vdots & & \vdots & \vdots & \vdots \\
1 & a_n & a_n^2 & \cdots & a_n^{n-3} & a_n^{n-1} & a_n^n
\end{array}\right|.\tiny\boxed{\begin{array}{c}\mbox{跟锦数学微信公众号}\\\mbox{www.zhangzujin.cn}\end{array}}\end{aligned} $$
[例1.1.22(湘潭大学,2023)](https://mp.weixin.qq.com/s?__biz=MzU3OTExMzI1Mg==&mid=2247541955&idx=2&sn=d4b1afe292d9bf8f65ae911fbb0b8204&chksm=fd697c88ca1ef59e037f956bf0838d244a23a9edeee569c022772e05b310b8a6f4f80c461ceb)
### 递推法
递推法就是将 $\displaystyle n$ 阶行列式 $\displaystyle D_n$ 用 $\displaystyle D_{n-1}$ 或更低阶的行列式表示出来, 然后通过递推求出 $\displaystyle D_n$ .
例 1.1.23 (江苏大学,2004; 西南师范大学,2004; 沈阳工业大学,2018; 长沙理工大学,2020; 山东师范大学,2021; 北京工业大学,2021; 陕西师范大学,2022) 计算行列式
$\displaystyle \tiny\boxed{@跟锦数学微信公众号}$
$$ \begin{aligned} D_n=\left|\begin{array}{rrrrr}
x & a & a & \cdots & a \\
-a & x & a & \cdots & a \\
-a & -a & x & \cdots & a \\
\vdots & \vdots & \vdots & & \vdots \\
-a & -a & -a & \cdots & x
\end{array}\right|.\tiny\boxed{\begin{array}{c}\mbox{跟锦数学微信公众号}\\\mbox{www.zhangzujin.cn}\end{array}}\end{aligned} $$
[例1.1.23(江苏大学,2004;西南师范大学,2004;沈阳工业大学,2018;](https://mp.weixin.qq.com/s?__biz=MzU3OTExMzI1Mg==&mid=2247541955&idx=1&sn=f394d564431444c66915156e16e2455a&chksm=fd697c88ca1ef59ed722b9b0b93a49bd16e4bc4fe1eba4bd40f0108edf05d9598b7b0ebde793)
例 1.1.24 (东北师范大学,2016; 河北工业大学,2020) 证明:
$\displaystyle \tiny\boxed{@跟锦数学微信公众号}$
$$ \begin{aligned} \left|\begin{array}{ccccc}
x & y & \cdots & y & y \\
z & x & \cdots & y & y \\
\vdots & \vdots & & \vdots & \vdots \\
z & z & \cdots & x & y \\
z & z & \cdots & z & x
\end{array}\right|=\frac{y(x-z)^n-z(x-y)^n}{y-z}(y \neq z).\tiny\boxed{\begin{array}{c}\mbox{跟锦数学微信公众号}\\\mbox{www.zhangzujin.cn}\end{array}}\end{aligned} $$
[例1.1.24(东北师范大学,2016;河北工业大学,2020)](https://mp.weixin.qq.com/s?__biz=MzU3OTExMzI1Mg==&mid=2247541967&idx=6&sn=60f4dda7ba46f69ff3368477d9bed8b2&chksm=fd697c84ca1ef592431fde7ddf905f70d4edb4e596db657695931bc90d8d9b9d5419d8bb6056)
例 1.1.25 (兰州大学,2010) 计算下列行列式的值:
$\displaystyle \tiny\boxed{@跟锦数学微信公众号}$
$$ \begin{aligned} D_n=\left|\begin{array}{ccccc}
x & b & \cdots & b & b \\
a & x & \cdots & b & b \\
\vdots & \vdots & & \vdots & \vdots \\
a & a & \cdots & x & b \\
a & a & \cdots & a & x
\end{array}\right|.\tiny\boxed{\begin{array}{c}\mbox{跟锦数学微信公众号}\\\mbox{www.zhangzujin.cn}\end{array}}\end{aligned} $$
[例1.1.25(兰州大学,2010)](https://mp.weixin.qq.com/s?__biz=MzU3OTExMzI1Mg==&mid=2247541967&idx=5&sn=069ac6d6a90e53c0b5f29c4c251616cc&chksm=fd697c84ca1ef592ed90e7265578f8e5051a668fcd972acde2eb35b959b44865050024149792)
例 1.1.26 (西安建筑科技大学, 2020; 武汉理工大学,2021) 计算 $\displaystyle n$ 阶行列式
$\displaystyle \tiny\boxed{@跟锦数学微信公众号}$
$$ \begin{aligned} D_n=\left|\begin{array}{cccccc}
1 & 2 & 3 & \cdots & n-1 & n \\
x & 1 & 2 & \cdots & n-2 & n-1 \\
x & x & 1 & \cdots & n-3 & n-2 \\
\vdots & \vdots & \vdots & & \vdots & \vdots \\
x & x & x & \cdots & 1 & 2 \\
x & x & x & \cdots & x & 1
\end{array}\right|.\tiny\boxed{\begin{array}{c}\mbox{跟锦数学微信公众号}\\\mbox{www.zhangzujin.cn}\end{array}}\end{aligned} $$
[例1.1.26(西安建筑科技大学,2020;武汉理工大学,2021)](https://mp.weixin.qq.com/s?__biz=MzU3OTExMzI1Mg==&mid=2247541967&idx=4&sn=a32e5276afcc8890caf2f89a0e72ffc5&chksm=fd697c84ca1ef5921a213b68ce4f06d600e805b4266a0f96e2720eb1d12f5c3f6dce555fd008)
例 1.1.27 (赣南师范大学, 2017; 山东科技大学,2020) 计算 $\displaystyle n$ 阶行列式
$\displaystyle \tiny\boxed{@跟锦数学微信公众号}$
$$ \begin{aligned} D_n=\left|\begin{array}{cccc}
\alpha+\beta & \alpha \beta & & \\
1 & \alpha+\beta & \ddots & \\
& \ddots & \ddots & \alpha \beta \\
& & 1 & \alpha+\beta
\end{array}\right|.\tiny\boxed{\begin{array}{c}\mbox{跟锦数学微信公众号}\\\mbox{www.zhangzujin.cn}\end{array}}\end{aligned} $$
[例1.1.27(赣南师范大学,2017;山东科技大学,2020)](https://mp.weixin.qq.com/s?__biz=MzU3OTExMzI1Mg==&mid=2247541967&idx=3&sn=dda2ee915133e2bb8a85909ad843bae4&chksm=fd697c84ca1ef592f291d7bb8fbbd68a68d5c6408b5c42db7b43b73effa5cdb55184ed99558c)
例 1.1.28 (河北大学, 2014 ; 河北工业大学,2022) 计算 $\displaystyle n$ 阶行列式
$\displaystyle \tiny\boxed{@跟锦数学微信公众号}$
$$ \begin{aligned} \left|\begin{array}{cccccc}
2 a & a^2 & & & & \\
1 & 2 a & a^2 & & & \\
& 1 & 2 a & a^2 & & \\
& & \ddots & \ddots & \ddots & \\
& & & 1 & 2 a & a^2 \\
& & & & 1 & 2 a
\end{array}\right|.\tiny\boxed{\begin{array}{c}\mbox{跟锦数学微信公众号}\\\mbox{www.zhangzujin.cn}\end{array}}\end{aligned} $$
[例1.1.28(河北大学,2014;河北工业大学,2022)](https://mp.weixin.qq.com/s?__biz=MzU3OTExMzI1Mg==&mid=2247541967&idx=2&sn=aaaf7e93a8125ffac4e62da563b36f0b&chksm=fd697c84ca1ef592b11fc69e8f88d1550d439de2e928cd0f533f9e00eae086afc57f77577846)
例 1.1.29 (北京科技大学,2020) 计算下列行列式:
$\displaystyle \tiny\boxed{@跟锦数学微信公众号}$
$$ \begin{aligned} D_n=\left|\begin{array}{cccccc}
a^2+a b & a^2 b & 0 & \cdots & 0 & 0 \\
1 & a+b & a b & \cdots & 0 & 0 \\
0 & 1 & a+b & \cdots & 0 & 0 \\
\vdots & \vdots & \vdots & & \vdots & \vdots \\
0 & 0 & 0 & \cdots & a+b & a b \\
0 & 0 & 0 & \cdots & 1 & a+b
\end{array}\right|.\tiny\boxed{\begin{array}{c}\mbox{跟锦数学微信公众号}\\\mbox{www.zhangzujin.cn}\end{array}}\end{aligned} $$
[例1.1.29(北京科技大学,2020)](https://mp.weixin.qq.com/s?__biz=MzU3OTExMzI1Mg==&mid=2247541967&idx=1&sn=9b864ee159fb4d1296b343c6bcdf2d84&chksm=fd697c84ca1ef5926f737d81b30f7bdaa0e2d205a8196b3b257879ef6c301509c32a84705337)
例 1.1.30 计算 $\displaystyle n$ 阶行列式
$\displaystyle \tiny\boxed{@跟锦数学微信公众号}$
$$ \begin{aligned} D_n=\left|\begin{array}{cccccc}
a & b & 0 & \cdots & 0 & 0 \\
c & a & b & \cdots & 0 & 0 \\
0 & c & a & \cdots & 0 & 0 \\
\vdots & \vdots & \vdots & & \vdots & \vdots \\
0 & 0 & 0 & \cdots & a & b \\
0 & 0 & 0 & \cdots & c & a
\end{array}\right|.\tiny\boxed{\begin{array}{c}\mbox{跟锦数学微信公众号}\\\mbox{www.zhangzujin.cn}\end{array}}\end{aligned} $$
[例1.1.30](https://mp.weixin.qq.com/s?__biz=MzU3OTExMzI1Mg==&mid=2247542031&idx=5&sn=a94c60294ff39bf792699dd6ebe3a909&chksm=fd697d44ca1ef45269803dcb44f17400dae6296731ef681a5d71cdf8cfca93475d4abe78136f)
例 1.1.31 (汕头大学, 2014) 计算下列矩阵的行列式 $\displaystyle \left(A_n\right.$ 的 $\displaystyle (i, n-i+1)$ 元素为 $\displaystyle a_i$ , 其他元素为 $\displaystyle 0 ; B_n$ 为三对角矩阵):
$\displaystyle \tiny\boxed{@跟锦数学微信公众号}$
$$ \begin{aligned} \begin{array}{c}
A_n=\left(\begin{array}{ccccc}
0 & 0 & \cdots & 0 & a_1 \\
0 & 0 & \cdots & a_2 & 0 \\
\vdots & \vdots & & \vdots & \vdots \\
0 & a_{n-1} & \cdots & 0 & 0 \\
a_n & 0 & \cdots & 0 & 0
\end{array}\right) ; \\
B_n=\left(\begin{array}{cccccc}
a_1 & b_1 & & & & \\
-a_1 & a_2-b_1 & b_2 & & & \\
& -a_2 & a_3-b_2 & & & \\
& & \ddots & \ddots & & \\
& & & & a_{n-1}-b_{n-2} & b_{n-1} \\
& & & & -a_{n-1} & a_n-b_{n-1}
\end{array}\right).
\end{array}\tiny\boxed{\begin{array}{c}\mbox{跟锦数学微信公众号}\\\mbox{www.zhangzujin.cn}\end{array}}\end{aligned} $$
[例1.1.31(汕头大学,2014)](https://mp.weixin.qq.com/s?__biz=MzU3OTExMzI1Mg==&mid=2247542031&idx=4&sn=96b29ba58f8188eaed26a87e5fd12293&chksm=fd697d44ca1ef452d2513e29c022266cb9b6904129ce6e3cf89b97b1ff227f3b3f4fd3455505)
例 1.1.32 (华中科技大学,2010) 计算 $\displaystyle n$ 阶行列式
$\displaystyle \tiny\boxed{@跟锦数学微信公众号}$
$$ \begin{aligned} \left|\begin{array}{ccccc}
1-x & x & & & \\
-1 & 1-x & x & & \\
& \ddots & \ddots & \ddots & \\
& & -1 & 1-x & x \\
& & & -1 & 1-x
\end{array}\right|.\tiny\boxed{\begin{array}{c}\mbox{跟锦数学微信公众号}\\\mbox{www.zhangzujin.cn}\end{array}}\end{aligned} $$
[例1.1.32(华中科技大学,2010)](https://mp.weixin.qq.com/s?__biz=MzU3OTExMzI1Mg==&mid=2247542031&idx=3&sn=5d3658d478d365ce8af72a8ff346807f&chksm=fd697d44ca1ef452ecc65e363e4428263eb22c7d4117711359f943b42bb22d771600e9c64a78)
例 1.1.33 (中国科学院大学,2017) 求下列 $\displaystyle n$ 阶行列式的值:
$\displaystyle \tiny\boxed{@跟锦数学微信公众号}$
$$ \begin{aligned} D_n=\left|\begin{array}{ccccccc}
1-a_1 & a_2 & 0 & 0 & \cdots & 0 & 0 \\
-1 & 1-a_2 & a_3 & 0 & \cdots & 0 & 0 \\
0 & -1 & 1-a_3 & a_4 & \cdots & 0 & 0 \\
\vdots & \vdots & \vdots & \vdots & & \vdots & \vdots \\
0 & 0 & 0 & 0 & \cdots & 1-a_{n-1} & a_n \\
0 & 0 & 0 & 0 & \cdots & -1 & 1-a_n
\end{array}\right|.\tiny\boxed{\begin{array}{c}\mbox{跟锦数学微信公众号}\\\mbox{www.zhangzujin.cn}\end{array}}\end{aligned} $$
[例1.1.33(中国科学院大学,2017)](https://mp.weixin.qq.com/s?__biz=MzU3OTExMzI1Mg==&mid=2247542031&idx=2&sn=dc81a9278ad071a765fc8591202b5914&chksm=fd697d44ca1ef452db7411f4b705d1ac4bd8e994df5892708c6b6da3a0f74b3efeaaea43bb49)
### 利用已知行列式
利用范德蒙德行列式等的结论计算.
例 1.1.34 (南开大学,2022) 计算行列式
$\displaystyle \tiny\boxed{@跟锦数学微信公众号}$
$$ \begin{aligned} \left|\begin{array}{llll}
2^4+1 & 2^3 & 2^2 & 2 \\
3^4+1 & 3^3 & 3^2 & 3 \\
4^4+1 & 4^3 & 4^2 & 4 \\
5^4+1 & 5^3 & 5^2 & 5
\end{array}\right|.\tiny\boxed{\begin{array}{c}\mbox{跟锦数学微信公众号}\\\mbox{www.zhangzujin.cn}\end{array}}\end{aligned} $$
[例1.1.34(南开大学,2022)](https://mp.weixin.qq.com/s?__biz=MzU3OTExMzI1Mg==&mid=2247542031&idx=1&sn=a60f816f8f0f897e9817f6c8e3fc5b43&chksm=fd697d44ca1ef4524b9de96c68e8bd47827a30d7106a010f6761370801a7870ff5e2582a6dab)
例 1.1.35 (西南大学,2019) 计算行列式
$\displaystyle \tiny\boxed{@跟锦数学微信公众号}$
$$ \begin{aligned} \left|\begin{array}{cccc}
1 & 1 & 1 & 1 \\
1 & 2 & 3 & 4 \\
1 & 4 & 9 & 16 \\
1 & 8 & 27 & 256
\end{array}\right|.\tiny\boxed{\begin{array}{c}\mbox{跟锦数学微信公众号}\\\mbox{www.zhangzujin.cn}\end{array}}\end{aligned} $$
[例1.1.35(西南大学,2019)](https://mp.weixin.qq.com/s?__biz=MzU3OTExMzI1Mg==&mid=2247542075&idx=7&sn=d1608996e37d391c7aedfd5e1d251817&chksm=fd697d70ca1ef4665b9e0bc1acf6b046bbfc8309999158447c8b464b8f8e7a33f69f064494ac)
例 1.1.36 (湖北大学,2000) 设
$\displaystyle \tiny\boxed{@跟锦数学微信公众号}$
$$ \begin{aligned} V=\left|\begin{array}{ccccc}
1 & 1 & \cdots & 1 & 1 \\
1 & 2 & \cdots & 19 & 20 \\
1 & 2^2 & \cdots & 19^2 & 20^2 \\
\vdots & \vdots & & \vdots & \vdots \\
1 & 2^{19} & \cdots & 19^{19} & 20^{19}
\end{array}\right|.\tiny\boxed{\begin{array}{c}\mbox{跟锦数学微信公众号}\\\mbox{www.zhangzujin.cn}\end{array}}\end{aligned} $$
(1) 求 $\displaystyle V$ 写成阶乘形式的值;
(2) $\displaystyle V$ 的值的末尾有多少个零?
[例1.1.36(湖北大学,2000)](https://mp.weixin.qq.com/s?__biz=MzU3OTExMzI1Mg==&mid=2247542075&idx=6&sn=d05dc21b5f3fc692aee7665bf7466349&chksm=fd697d70ca1ef466a650187f468671159d703517a37e21642981cb7e7390d1841fcb349234f6)
例 1.1.37 (福州大学,2006; 河北工业大学,2006; 北京交通大学,2007) 已知行列式
$\displaystyle \tiny\boxed{@跟锦数学微信公众号}$
$$ \begin{aligned} P(x)=\left|\begin{array}{ccccc}
1 & x & x^2 & \cdots & x^{n-1} \\
1 & a_1 & a_1^2 & \cdots & a_1^{n-1} \\
1 & a_2 & a_2^2 & \cdots & a_2^{n-1} \\
\vdots & \vdots & \vdots & & \vdots \\
1 & a_{n-1} & a_{n-1}^2 & \cdots & a_{n-1}^{n-1}
\end{array}\right|,\tiny\boxed{\begin{array}{c}\mbox{跟锦数学微信公众号}\\\mbox{www.zhangzujin.cn}\end{array}}\end{aligned} $$
其中 $\displaystyle a_1, a_2, \cdots, a_{n-1}$ 为互不相同的数. 证明:$\displaystyle P(x)$ 是一个 $\displaystyle n-1$ 次多项式, 并求其最高次项的系数和 $\displaystyle P(x)$ 的根.
[例1.1.37(福州大学,2006;?河北工业大学,2006;?北京交通大学,2007)](https://mp.weixin.qq.com/s?__biz=MzU3OTExMzI1Mg==&mid=2247542075&idx=5&sn=d3f16847e5b5013527c574e62d547014&chksm=fd697d70ca1ef466776fa9f37f51c79718f087b17500d6cf6a89abda3ff40d29f7e4d40f104b)
例 1.1.38 (南开大学,2005年) 计算行列式
$\displaystyle \tiny\boxed{@跟锦数学微信公众号}$
$$ \begin{aligned} D_n=\left|\begin{array}{cccc}
1 & 1 & \cdots & 1 \\
x_1+1 & x_2+1 & \cdots & x_n+1 \\
x_1^2+x_1 & x_2^2+x_2 & \cdots & x_n^2+x_n \\
x_1^3+x_1^2 & x_2^3+x_2^2 & \cdots & x_n^3+x_n^2 \\
\vdots & \vdots & & \vdots \\
x_1^{n-1}+x_1^{n-2} & x_2^{n-1}+x_2^{n-1} & \cdots & x_n^{n-1}+x_n^{n-2}
\end{array}\right|.\tiny\boxed{\begin{array}{c}\mbox{跟锦数学微信公众号}\\\mbox{www.zhangzujin.cn}\end{array}}\end{aligned} $$
[例1.1.38(南开大学,2005年)](https://mp.weixin.qq.com/s?__biz=MzU3OTExMzI1Mg==&mid=2247542075&idx=4&sn=7661d02d808b85655505dbbee06cd58b&chksm=fd697d70ca1ef466df8a5bd30599bffc377b5954ec8146c2cfc88960574cca68a44b3d453695)
例 1.1.39 (深圳大学,2013; 西北大学,2014; 聊城大学,2015) 设 $\displaystyle a_i \neq 0(i=1,2, \cdots, n+1)$ . 计算 $\displaystyle n+1$ 阶行列式
$\displaystyle \tiny\boxed{@跟锦数学微信公众号}$
$$ \begin{aligned} D=\left|\begin{array}{ccccc}
a_1^n & a_1^{n-1} b_1 & \ldots & a_1 b_1^{n-1} & b_1^n \\
a_2^n & a_2^{n-1} b_2 & \ldots & a_2 b_2^{n-1} & b_2^n \\
\vdots & \vdots & & \vdots & \vdots \\
a_{n+1}^n & a_{n+1}^{n-1} b_{n+1} & \ldots & a_{n+1} b_{n+1}^{n-1} & b_{n+1}^n
\end{array}\right|.\tiny\boxed{\begin{array}{c}\mbox{跟锦数学微信公众号}\\\mbox{www.zhangzujin.cn}\end{array}}\end{aligned} $$
[例1.1.39(深圳大学,2013;?西北大学,2014;?聊城大学,2015)](https://mp.weixin.qq.com/s?__biz=MzU3OTExMzI1Mg==&mid=2247542075&idx=3&sn=b0bcd371fe1742d4a380352542d3f6ff&chksm=fd697d70ca1ef466c9f6b7307b78d182ca898528eba7016d77afee53c8b648a711369171164b)
例 1.1.40 (扬州大学, 2019) 设 $\displaystyle S_k=x_1^k+x_2^k+\cdots+x_n^k(k=0,1,2,3, \cdots)$ , 证明:
(1) $\displaystyle n=3$ 时, 行列式 $\displaystyle D_3=\left|\begin{array}{lll}S_0 & S_1 & S_2 \\ S_1 & S_2 & S_3 \\ S_2 & S_3 & S_4\end{array}\right|=\prod_{1 \leqslant i\lt j \leqslant 3}\left(x_j-x_i\right)^2$ ;
(2) $\displaystyle n+1$ 阶行列式
$\displaystyle \tiny\boxed{@跟锦数学微信公众号}$
$$ \begin{aligned}
D_{n+1}=\left|\begin{array}{ccccc}S_0 & S_1 & \cdots & S_{n-1} & 1 \\ S_1 & S_2 & \cdots & S_n & x \\ S_2 & S_3 & \cdots & S_{n+1} & x^2 \\ \vdots & \vdots & & \vdots & \vdots \\ S_n & S_{n+1} & \cdots & S_{2 n-1} & x^n\end{array}\right|=\prod_{1 \leqslant i\lt j \leqslant n}\left(x_j-x_i\right)^2 \prod_{i=1}^n\left(x-x_i\right).
\tiny\boxed{\begin{array}{c}\mbox{跟锦数学微信公众号}\\\mbox{www.zhangzujin.cn}\end{array}}\end{aligned} $$
[例 1.1.40 (扬州大学, 2019)](https://mp.weixin.qq.com/s?__biz=MzU3OTExMzI1Mg==&mid=2247542142&idx=6&sn=7018d9b9a02fde335da70d82fb74dab6&chksm=fd697d35ca1ef42323aecd7186e162e9d77a261b7aa1033f3808facfac490efa49fb0d5bb8ce)
例 1.1.41 (四川大学,2011) 设 $\displaystyle F, K$ 都是数域且 $\displaystyle F \subseteq K$ . 设 $\displaystyle F$ 上的 $\displaystyle n$ 次多项式 $\displaystyle f(x)$ 在 $\displaystyle K$ 上有 $\displaystyle n$ 个根 $\displaystyle x_1, x_2, \cdots, x_n$ , 证明:$\displaystyle \prod_{1 \leqslant i\lt j \leqslant n}\left(x_i-x_j\right)^2 \in F$ .
[例1.1.41(四川大学,2011)](https://mp.weixin.qq.com/s?__biz=MzU3OTExMzI1Mg==&mid=2247542090&idx=7&sn=29b3bedbe550d99160840eaef4e88eae&chksm=fd697d01ca1ef4178cbff029949c566277d5eac4da4a07f31c39799461cd4a1cb5264306c565)
例 1.1.42 (兰州大学,2010; 兰州大学,2015; 兰州大学,2020; 杭州师范大学,2020) 计算行列式
$\displaystyle \tiny\boxed{@跟锦数学微信公众号}$
$$ \begin{aligned} D_n=\left|\begin{array}{cccc}
1+x_1 & 1+x_1^2 & \cdots & 1+x_1^n \\
1+x_2 & 1+x_2^2 & \cdots & 1+x_2^n \\
\vdots & \vdots & & \vdots \\
1+x_n & 1+x_n^2 & \cdots & 1+x_n^n
\end{array}\right|.\tiny\boxed{\begin{array}{c}\mbox{跟锦数学微信公众号}\\\mbox{www.zhangzujin.cn}\end{array}}\end{aligned} $$
[例1.1.42(兰州大学,2010;兰州大学,2015;兰州大学,2020;](https://mp.weixin.qq.com/s?__biz=MzU3OTExMzI1Mg==&mid=2247542090&idx=6&sn=ead1340bd7b524ca61e39baab0ed9c7f&chksm=fd697d01ca1ef417c39c666fdb7caf7beb7f49ebaf2def56888976b6648bccd4a13f61660f70)
例 1.1.43 (兰州大学,2021) 计算行列式
$\displaystyle \tiny\boxed{@跟锦数学微信公众号}$
$$ \begin{aligned} D_n=\left|\begin{array}{cccc}
a+x_1 & a+x_2 & \cdots & a+x_n \\
a+x_1^2 & a+x_2^2 & \cdots & a+x_n^2 \\
\vdots & \vdots & & \vdots \\
a+x_1^n & a+x_2^n & \cdots & a+x_n^n
\end{array}\right|.\tiny\boxed{\begin{array}{c}\mbox{跟锦数学微信公众号}\\\mbox{www.zhangzujin.cn}\end{array}}\end{aligned} $$
[例1.1.43(兰州大学,2021)](https://mp.weixin.qq.com/s?__biz=MzU3OTExMzI1Mg==&mid=2247542090&idx=5&sn=3245a1a2ebf8f310b761a45b542ce1b2&chksm=fd697d01ca1ef41703e0b99796f0436af0102d854c86cdcdec2fb5f0988d12fd67c54c27f843)
例 1.1.44 (陕西师范大学, 2021; 兰州大学,2021; 北京邮电大学,2022) 计算如下 $\displaystyle n$ 阶行列式:
$\displaystyle \tiny\boxed{@跟锦数学微信公众号}$
$$ \begin{aligned} D_n=\left|\begin{array}{cccc}
a+x_1 & a+x_2 & \cdots & a+x_n \\
a^2+x_1^2 & a^2+x_2^2 & \cdots & a^2+x_n^2 \\
\vdots & \vdots & & \vdots \\
a^n+x_1^n & a^n+x_2^n & \cdots & a^n+x_n^n
\end{array}\right|.\tiny\boxed{\begin{array}{c}\mbox{跟锦数学微信公众号}\\\mbox{www.zhangzujin.cn}\end{array}}\end{aligned} $$
[例1.1.44(陕西师范大学,2021;兰州大学,2021;北京邮电大学,2022)](https://mp.weixin.qq.com/s?__biz=MzU3OTExMzI1Mg==&mid=2247542090&idx=4&sn=bf37c5f9d9f02597709eb8aa62be462f&chksm=fd697d01ca1ef417d02560279f5b63e1bca32172520002e2e4f022764ffbde9edcce2d60a2c4)
例 1.1.45 (华中科技大学,2023) 计算行列式
$\displaystyle \tiny\boxed{@跟锦数学微信公众号}$
$$ \begin{aligned} \left|\begin{array}{cccc}
2^2-2 & 2^3-2 & \cdots & 2^{2023}-2 \\
3^2-3 & 3^3-3 & \cdots & 3^{2023}-3 \\
\vdots & \vdots & & \vdots \\
2023^2-2023 & 2023^3-2023 & \cdots & 2023^{2023}-2023
\end{array}\right|.\tiny\boxed{\begin{array}{c}\mbox{跟锦数学微信公众号}\\\mbox{www.zhangzujin.cn}\end{array}}\end{aligned} $$
[例1.1.45(华中科技大学,2023)](https://mp.weixin.qq.com/s?__biz=MzU3OTExMzI1Mg==&mid=2247542090&idx=3&sn=cee14f0ad3867259a9c60858388a6336&chksm=fd697d01ca1ef4172c2f5b9021c54daaa8a5109ae09f644c03893bdbf75c7a872ed7cf4966da)
### 数学归纳法
例 1.1.46 计算如下行列式:
$\displaystyle \tiny\boxed{@跟锦数学微信公众号}$
$$ \begin{aligned} D_n=\left|\begin{array}{cccc}
x^2+1 & x & & \\
x & x^2+1 & \ddots & \\
& \ddots & \ddots & x \\
& & x & x^2+1
\end{array}\right|.\tiny\boxed{\begin{array}{c}\mbox{跟锦数学微信公众号}\\\mbox{www.zhangzujin.cn}\end{array}}\end{aligned} $$
[例1.1.46](https://mp.weixin.qq.com/s?__biz=MzU3OTExMzI1Mg==&mid=2247542090&idx=2&sn=7545bdce701af8f96193b1b8d706b17c&chksm=fd697d01ca1ef41789bc43268558bb73c69102b1c1f148d475f75f7e6a83c2964a845f06a6bf)
例 1.1.47 (复旦大学高等代数每周一题 [问题 2017A03]) 求下列 $\displaystyle n$ 阶行列式的值:
$\displaystyle \tiny\boxed{@跟锦数学微信公众号}$
$$ \begin{aligned} |A|=\left|\begin{array}{cccccc}
1+x^2 & x & 0 & \cdots & 0 & 0 \\
x+x^2 & 1+x^2 & x & \cdots & 0 & 0 \\
0 & x & 1+x^2 & \cdots & 0 & 0 \\
\vdots & \vdots & \vdots & & \vdots & \vdots \\
0 & 0 & 0 & \cdots & 1+x^2 & x \\
0 & 0 & 0 & \cdots & x & 1+x^2
\end{array}\right|.\tiny\boxed{\begin{array}{c}\mbox{跟锦数学微信公众号}\\\mbox{www.zhangzujin.cn}\end{array}}\end{aligned} $$
[例 1.1.47 (复旦大学高等代数每周一题 [问题 2017A03])](https://mp.weixin.qq.com/s?__biz=MzU3OTExMzI1Mg==&mid=2247542179&idx=7&sn=24e0a2eada3280416350548cb1f4113a&chksm=fd697de8ca1ef4fe3eacde1f10ca00c7e76ef2befc51deeaf5729722822991773a4238f82c11)
例 1.1.48 证明:如果 $\displaystyle n$ 阶行列式 $\displaystyle D=\left(a_{i j}\right)$ 中的所有元素都是 1 或 -1, 则当 $\displaystyle n \geqslant 3$ 时, $\displaystyle |D| \leqslant \frac{2}{3} n !$ .
[例 1.1.48](https://mp.weixin.qq.com/s?__biz=MzU3OTExMzI1Mg==&mid=2247542179&idx=6&sn=3c094fb853c2cff020c49aa01723fc1f&chksm=fd697de8ca1ef4fec2e0e9a9e214009677702fc61530cbb16a4f07baf85560ae2e801f2cbdab)
例 1.1.49 (第四届全国大学生数学竞赛预赛,2013) 设 $\displaystyle n$ 阶实方阵 $\displaystyle A$ 的每个元素的绝对值为 2. 证明:当 $\displaystyle n \geqslant 3$ 时, $\displaystyle |A| \leqslant \frac{1}{3} 2^{n+1} n !$ .
[例 1.1.49 (第四届全国大学生数学竞赛预赛,2013)](https://mp.weixin.qq.com/s?__biz=MzU3OTExMzI1Mg==&mid=2247542179&idx=5&sn=3cb521af88231c5972beb87ff4a08c86&chksm=fd697de8ca1ef4fe04ab500ad460f181a4263238e162339036332cedea2d2b52acfb1220cbbb)
例 1.1.50 (第十三届全国大学生数学竞赛预赛数学 B 类,2021) 设 $\displaystyle R=\{0,1,-1\}, S$ 为 $\displaystyle R$ 上的 3 阶行列式的全体, 即 $\displaystyle S=\left\{\operatorname{det}\left(a_{i j}\right)_{3 \times 3} \mid a_{i j} \in R\right\}$ . 证明:$\displaystyle S=\{-4,-3,-2,-1,0,1,2,3,4\}$ .
[例 1.1.50 (第十三届全国大学生数学竞赛预赛数学 B 类,2021)](https://mp.weixin.qq.com/s?__biz=MzU3OTExMzI1Mg==&mid=2247542179&idx=4&sn=9dbf82f52690d5a928937705b65ec364&chksm=fd697de8ca1ef4fe609a35c42347d7a280a39436a153c4555eb322972129a5241ad1d884db55)
### 定义法
例 1.1.51 证明:
$\displaystyle \tiny\boxed{@跟锦数学微信公众号}$
$$ \begin{aligned} \left|\begin{array}{cccccc}
1 & 2 & 3 & \cdots & n-1 & n \\
2^2 & 3^2 & 4^2 & \cdots & n^2 & (n+1)^2 \\
3^3 & 4^3 & 5^3 & \cdots & (n+1)^3 & (n+1)^3 \\
\vdots & \vdots & \vdots & & \vdots & \vdots \\
n^n & (n+1)^n & (n+1)^n & \cdots & (n+1)^n & (n+1)^n
\end{array}\right| \neq 0\tiny\boxed{\begin{array}{c}\mbox{跟锦数学微信公众号}\\\mbox{www.zhangzujin.cn}\end{array}}\end{aligned} $$
其中 $\displaystyle n$ 为奇数.
[例 1.1.51](https://mp.weixin.qq.com/s?__biz=MzU3OTExMzI1Mg==&mid=2247542179&idx=3&sn=8c2a1c82c2734504b5b7299e79cf5b1d&chksm=fd697de8ca1ef4fee3b1304d0d38c4bd28f690ec27c8c94824b8c2cb2ad5a7a4a2f9d5778a4e)
例 1.1.52 (樊启斌高等代数典型问题与方法思考与练习3.24) 求解齐次线性方程组
$\displaystyle \tiny\boxed{@跟锦数学微信公众号}$
$$ \begin{aligned} \left\{\begin{array}{c}
a_{11} x_1+a_{12} x_2+\cdots+a_{1 n} x_n=0, \\
a_{21} x_1+a_{22} x_2+\cdots+a_{2 n} x_n=0, \\
\vdots \\
a_{n 1} x_1+a_{n 2} x_2+\cdots+a_{n n} x_n=0,
\end{array}\right.\tiny\boxed{\begin{array}{c}\mbox{跟锦数学微信公众号}\\\mbox{www.zhangzujin.cn}\end{array}}\end{aligned} $$
其中 $\displaystyle a_{i i}=2019, i=1,2, \cdots, n, a_{i j} \in\{2018,610,-2018,-610\}, i \neq j(i, j=1,2, \cdots, n)$ .
[例 1.1.52 (樊启斌高等代数典型问题与方法思考与练习3.24)](https://mp.weixin.qq.com/s?__biz=MzU3OTExMzI1Mg==&mid=2247542179&idx=2&sn=208e692eb928c2fc15ecdf0fc18e7859&chksm=fd697de8ca1ef4fe6c87979441765c05eab3a59310b6bc87348187edfffeba7e341ab71d5320)
例 1.1.53 (武汉大学, 2020 $\displaystyle )$ 设 $\displaystyle A=\left(a_{i j}\right)_{n \times n}, a_{i j} \in \mathbb{Z}(i, j=1,2, \cdots, n), k$ 为正整数且 $\displaystyle k \geqslant 2$ .证明:
$\displaystyle \tiny\boxed{@跟锦数学微信公众号}$
$$ \begin{aligned} \left|\begin{array}{ccccc}
a_{11}-\frac{1}{k} & a_{12} & a_{13} & \cdots & a_{1 n} \\
a_{21} & a_{22}-\frac{1}{k} & a_{23} & \cdots & a_{2 n} \\
\vdots & \vdots & \vdots & & \vdots \\
a_{n 1} & a_{n 2} & a_{n 3} & \cdots & a_{n n}-\frac{1}{k}
\end{array}\right| \neq 0.\tiny\boxed{\begin{array}{c}\mbox{跟锦数学微信公众号}\\\mbox{www.zhangzujin.cn}\end{array}}\end{aligned} $$
[例 1.1.53 (武汉大学, 2020)](https://mp.weixin.qq.com/s?__biz=MzU3OTExMzI1Mg==&mid=2247542180&idx=6&sn=134c1b98a481b63c800b4678109be94a&chksm=fd697defca1ef4f91614edcb8400bb3579f092660bd88990a5770c03777ff88577a8e7803c27)
例 1.1.54 (西北大学,2020) 已知 $\displaystyle a_{i j}$ 都是整数, 证明:
$\displaystyle \tiny\boxed{@跟锦数学微信公众号}$
$$ \begin{aligned} \left|\begin{array}{ccccc}
a_{11}-\frac{1}{2} & a_{12} & a_{13} & \cdots & a_{1 n} \\
a_{21} & a_{22}-\frac{1}{2} & a_{23} & \cdots & a_{2 n} \\
\vdots & \vdots & \vdots & & \vdots \\
a_{n 1} & a_{n 2} & a_{n 3} & \cdots & a_{n n}-\frac{1}{2}
\end{array}\right| \neq 0.\tiny\boxed{\begin{array}{c}\mbox{跟锦数学微信公众号}\\\mbox{www.zhangzujin.cn}\end{array}}\end{aligned} $$
[例 1.1.54 (西北大学,2020)](https://mp.weixin.qq.com/s?__biz=MzU3OTExMzI1Mg==&mid=2247542180&idx=5&sn=7d783750d3d9281c5afd964be63721b1&chksm=fd697defca1ef4f90ac8c924722adc7cad7454eaf4c117b63c924e8ee221e3b97b75d320e67e)
例 1.1.55 (河南师范大学, 2015) 设 $\displaystyle A$ 是整数方阵, 证明:线性方程组 $\displaystyle A X=\frac{1}{2} X$ 只有零解.
[例 1.1.55 (河南师范大学, 2015)](https://mp.weixin.qq.com/s?__biz=MzU3OTExMzI1Mg==&mid=2247542180&idx=4&sn=4852d03fa0c759dfcd2e94c1eb222fd7&chksm=fd697defca1ef4f9833d7f6d83941c9c47b59f3717a650f1133e690b8d94a7749e86c7b66e57)
例 1.1.56 (西安交通大学,2008) 设 $\displaystyle n$ 阶方阵 $\displaystyle A$ 的元素都是整数, 有理数 $\displaystyle b=\frac{q}{p}$ 为既约分数 (即 $\displaystyle p \neq 1$ , 且 $\displaystyle p, q$ 互质), 证明:线性方程组 $\displaystyle A x=b x$ 只有零解.
[例 1.1.56 (西安交通大学,2008)](https://mp.weixin.qq.com/s?__biz=MzU3OTExMzI1Mg==&mid=2247542180&idx=3&sn=80c17c6c33561f551af64c7774d1b485&chksm=fd697defca1ef4f97341a91134225741443d161392eab0d603086832642eab34e84eba9887f1)
例 1.1.57 设 $\displaystyle A$ 是 $\displaystyle n$ 阶整数矩阵, 证明:线性方程组 $\displaystyle A X=B X$ 只有零解, 其中 $\displaystyle B=\operatorname{diag}$$\displaystyle \left(\frac{1}{2}, \frac{1}{2^2}, \cdots, \frac{1}{2^n}\right)$ .
[例 1.1.57](https://mp.weixin.qq.com/s?__biz=MzU3OTExMzI1Mg==&mid=2247542180&idx=2&sn=8d9f15f07a85ba7536cff6fe30861a09&chksm=fd697defca1ef4f9675998c79d693d5a523882197abb8df46a748be977d922d3918384c8b60b)
## 行列式的计算公式51题
常用的行列式计算公式
设 $\displaystyle A, B \in F^{n \times n}, k \in F$ , 则
(1) 一般情况下, $\displaystyle |A \pm B| \neq|A| \pm|B|$ ;
(2) $\displaystyle |k A|=k^n|A|$
(3) $\displaystyle |A B|=|A||B| ;\left|A^k\right|=|A|^k$ ;
(4) $\displaystyle \left|A^{\mathrm{T}}\right|=|A|$ ;
(5) $\displaystyle A$ 可逆的充要条件是 $\displaystyle |A| \neq 0$ ;
(6) 若 $\displaystyle A$ 可逆, 则 $\displaystyle \left|A^{-1}\right|=|A|^{-1}$ ;
(7) $\displaystyle \left|A^\star \right|=|A|^{n-1}$ ;
(8) 初等矩阵的行列式:$\displaystyle |P(i, j)|=-1,|P(i(c))|=c,|P(i, j(k))|=1$ ;
(9) 分块矩阵的行列式
$\displaystyle \tiny\boxed{@跟锦数学微信公众号}$
$$ \begin{aligned} \begin{array}{l}
\left|\begin{array}{ll}
A & O \\
O & B
\end{array}\right|=\left|\begin{array}{ll}
A & O \\
C & B
\end{array}\right|=\left|\begin{array}{ll}
A & D \\
O & B
\end{array}\right|=|A||B| ; \\
\left|\begin{array}{cc}
O & A_{n \times n} \\
B_{m \times m} & O
\end{array}\right|=\left|\begin{array}{cc}
C & A_{n \times n} \\
B_{m \times m} & O
\end{array}\right|=\left|\begin{array}{cc}
O & A_{n \times n} \\
B_{m \times m} & D
\end{array}\right|=(-1)^{m n}|A||B|. \\\end{array}\tiny\boxed{\begin{array}{c}\mbox{跟锦数学微信公众号}\\\mbox{www.zhangzujin.cn}\end{array}}\end{aligned} $$
例 1.2.1 (东北师范大学,2022) 计算行列式
$\displaystyle \tiny\boxed{@跟锦数学微信公众号}$
$$ \begin{aligned} D_n=\left|\begin{array}{cccc}
a_1+b_1 & a_1+b_2 & \cdots & a_1+b_n \\
a_2+b_1 & a_2+b_2 & \cdots & a_2+b_n \\
\vdots & \vdots & & \vdots \\
a_n+b_1 & a_n+b_2 & \cdots & a_n+b_n
\end{array}\right|.\tiny\boxed{\begin{array}{c}\mbox{跟锦数学微信公众号}\\\mbox{www.zhangzujin.cn}\end{array}}\end{aligned} $$
[例 1.2.01 (东北师范大学,2022)](https://mp.weixin.qq.com/s?__biz=MzU3OTExMzI1Mg==&mid=2247542181&idx=7&sn=273d3f39b98231d647da16d21d353eaa&chksm=fd697deeca1ef4f890bb83f12e823e48d2d70288bf9298b0f1bad629ad12275d24d746714a7f)
例 1.2.2 (杭州电子科技大学,2020) 计算行列式
$\displaystyle \tiny\boxed{@跟锦数学微信公众号}$
$$ \begin{aligned} D=\left|\begin{array}{cccc}
a_1 b_1+1 & a_1 b_2+2 & \cdots & a_1 b_n+n \\
a_2 b_1+1 & a_2 b_2+2 & \cdots & a_2 b_n+n \\
\vdots & \vdots & & \vdots \\
a_n b_1+1 & a_n b_2+2 & \cdots & a_n b_n+n
\end{array}\right|.\tiny\boxed{\begin{array}{c}\mbox{跟锦数学微信公众号}\\\mbox{www.zhangzujin.cn}\end{array}}\end{aligned} $$
[例 1.2.02 (杭州电子科技大学,2020)](https://mp.weixin.qq.com/s?__biz=MzU3OTExMzI1Mg==&mid=2247542181&idx=6&sn=1cb6a4a779ceea970498e6242562a903&chksm=fd697deeca1ef4f89572dd87fef6d998fcfdffcf06eff2ccba50b68aec54be3f679fdc42170c)
例 1.2.3 (兰州大学,2022) 计算 $\displaystyle n$ 阶行列式
$\displaystyle \tiny\boxed{@跟锦数学微信公众号}$
$$ \begin{aligned} \left|\begin{array}{ccccc}
x_1+a_1 b_1 & x_1+a_1 b_2 & x_1+a_1 b_3 & \cdots & x_1+a_1 b_n \\
x_2+a_2 b_1 & x_2+a_2 b_2 & x_2+a_2 b_3 & \cdots & x_2+a_2 b_n \\
x_3+a_3 b_1 & x_3+a_3 b_2 & x_3+a_3 b_3 & \cdots & x_3+a_3 b_n \\
\vdots & \vdots & \vdots & & \vdots \\
x_n+a_n b_1 & x_n+a_n b_2 & x_n+a_n b_3 & \cdots & x_n+a_n b_n
\end{array}\right|.\tiny\boxed{\begin{array}{c}\mbox{跟锦数学微信公众号}\\\mbox{www.zhangzujin.cn}\end{array}}\end{aligned} $$
[例 1.2.03 (兰州大学,2022)](https://mp.weixin.qq.com/s?__biz=MzU3OTExMzI1Mg==&mid=2247542181&idx=5&sn=270ab2f954c962ec1116313d3bf64b6c&chksm=fd697deeca1ef4f8657424649156323284324892547bde1e39f89c7a118adb9cd367007dc348)
例 1.2.4 设 $\displaystyle M=\left(\begin{array}{ll}A & B \\ C & D\end{array}\right)$ 为 $\displaystyle m+n$ 阶方阵, 其中 $\displaystyle A$ 为 $\displaystyle m$ 阶可逆方阵, 证明
$\displaystyle \tiny\boxed{@跟锦数学微信公众号}$
$$ \begin{aligned} \operatorname{det} M=\operatorname{det} A\left(D-C A ^ { - 1 } \boldsymbol { B } \right)\tiny\boxed{\begin{array}{c}\mbox{跟锦数学微信公众号}\\\mbox{www.zhangzujin.cn}\end{array}}\end{aligned} $$
[例 1.2.04](https://mp.weixin.qq.com/s?__biz=MzU3OTExMzI1Mg==&mid=2247542181&idx=4&sn=09081e88c618664b8fa1a9154d271dc2&chksm=fd697deeca1ef4f8d6eafc70437d1b93ac5555944f4dc7c19bec5c174aeb8ac7c7280674798a)
例 1.2.5 (重庆大学,2022) 设 $\displaystyle A, B, C, D$ 为 $\displaystyle n$ 阶方阵, 且 $\displaystyle |A| \neq 0, A=C A$ . 证明:
$\displaystyle \tiny\boxed{@跟锦数学微信公众号}$
$$ \begin{aligned} \left|\begin{array}{ll}
A & B \\
C & D
\end{array}\right|=|A D-C B| \text {. }\tiny\boxed{\begin{array}{c}\mbox{跟锦数学微信公众号}\\\mbox{www.zhangzujin.cn}\end{array}}\end{aligned} $$
[例 1.2.05 (重庆大学,2022)](https://mp.weixin.qq.com/s?__biz=MzU3OTExMzI1Mg==&mid=2247542181&idx=3&sn=6fd8c2bba49a325723e4b7d19c04ecd0&chksm=fd697deeca1ef4f8fc89c1346d7e730ce185938a6c3f665a0d852f8a9b6d08a9206b4e378beb)
例 1.2.6 (兰州大学,2015; 河北工业大学,2021; 天津大学,2022) 设 $\displaystyle A, B, C, D$ 均为 $\displaystyle n$ 阶方阵, 且 $\displaystyle A C=C A$ , 则
$\displaystyle \tiny\boxed{@跟锦数学微信公众号}$
$$ \begin{aligned} \left|\begin{array}{ll}
A & B \\
C & D
\end{array}\right|=|A D-C B|.\tiny\boxed{\begin{array}{c}\mbox{跟锦数学微信公众号}\\\mbox{www.zhangzujin.cn}\end{array}}\end{aligned} $$
[例 1.2.06 (兰州大学,2015; 河北工业大学,2021; 天津大学,2022)](https://mp.weixin.qq.com/s?__biz=MzU3OTExMzI1Mg==&mid=2247542181&idx=2&sn=4854afa1c3edfd75fab87e3fad806573&chksm=fd697deeca1ef4f858578809b478efa3e5b0c513a48892118f997dd6e39a947b5c12e60a6eec)
例 1.2.7 (首都师范大学,2016) 求行列式
$\displaystyle \tiny\boxed{@跟锦数学微信公众号}$
$$ \begin{aligned} \left|\begin{array}{llll}
a^2 & a b & a b & b^2 \\
a c & a d & b c & b d \\
a c & b c & a d & b d \\
c^2 & c d & c d & d^2
\end{array}\right|.\tiny\boxed{\begin{array}{c}\mbox{跟锦数学微信公众号}\\\mbox{www.zhangzujin.cn}\end{array}}\end{aligned} $$
[例 1.2.07 (首都师范大学,2016)](https://mp.weixin.qq.com/s?__biz=MzU3OTExMzI1Mg==&mid=2247542182&idx=6&sn=d51690a0e1697de5b2e3d7e53fcf6c93&chksm=fd697dedca1ef4fb99e20609f95d2c1e3afcdf90f083048abcd6b96251394a30af19d8b4c620)
例 1.2.8 (首都师范大学, 2015) 设 $\displaystyle A, B, C, D$ 为 $\displaystyle n$ 阶方阵, 其中 $\displaystyle D$ 为可逆矩阵, 且 $\displaystyle C D=$$\displaystyle D C$ . 证明:
$\displaystyle \tiny\boxed{@跟锦数学微信公众号}$
$$ \begin{aligned} \operatorname{det}\left(\begin{array}{ll}
A & B \\
C & D
\end{array}\right)=\operatorname{det}(A D-B C).\tiny\boxed{\begin{array}{c}\mbox{跟锦数学微信公众号}\\\mbox{www.zhangzujin.cn}\end{array}}\end{aligned} $$
[例 1.2.08 (首都师范大学, 2015)](https://mp.weixin.qq.com/s?__biz=MzU3OTExMzI1Mg==&mid=2247542182&idx=5&sn=2946022f31134918ad4bf4e54b451cc1&chksm=fd697dedca1ef4fb04a4ac6e723028ad422c3a167e20f69205a640f32a800fdc07f7e1c9d52c)
例 1.2.9 设 $\displaystyle A, D$ 分别为 $\displaystyle n$ 阶与 $\displaystyle m$ 阶方阵, 则
$\displaystyle \tiny\boxed{@跟锦数学微信公众号}$
$$ \begin{aligned} \left|\begin{array}{ll}
A & B \\
C & D
\end{array}\right|=\left\{\begin{array}{ll}
|A|\left|D-C A^{-1} B\right|, & A \text { 可逆; } \\
|D|\left|A-B D^{-1} C\right|, & D \text { 可逆. }
\end{array}\right.\tiny\boxed{\begin{array}{c}\mbox{跟锦数学微信公众号}\\\mbox{www.zhangzujin.cn}\end{array}}\end{aligned} $$
[例 1.2.09](https://mp.weixin.qq.com/s?__biz=MzU3OTExMzI1Mg==&mid=2247542182&idx=4&sn=aff095278b688198ae7f60b75b816034&chksm=fd697dedca1ef4fbaa97040ebdbab8275d8ca2ec06aea098f1d9aaac17096d910e1c8b85a1eb)
例 1.2.10 (南京师范大学,2016) 设矩阵 $\displaystyle A, C$ 分别为 $\displaystyle n$ 阶与 $\displaystyle m$ 阶可逆矩阵, $\displaystyle B, D$ 分别为 $\displaystyle n \times m$ 和 $\displaystyle m \times n$ 矩阵. 证明:
$\displaystyle \tiny\boxed{@跟锦数学微信公众号}$
$$ \begin{aligned} |C|\left|A-B C^{-1} D\right|=|A|\left|C-A^{-1} B\right|.\tiny\boxed{\begin{array}{c}\mbox{跟锦数学微信公众号}\\\mbox{www.zhangzujin.cn}\end{array}}\end{aligned} $$
[例 1.2.10 (南京师范大学,2016)](https://mp.weixin.qq.com/s?__biz=MzU3OTExMzI1Mg==&mid=2247542182&idx=3&sn=5ce6f8fb4da204a569dda0d20116746f&chksm=fd697dedca1ef4fbda4da74835fef1da358fba6ef2d69139b89ad63580fd321bdc195f59442b)
例 1.2.11 (南京师范大学, 2020) 设 $\displaystyle A, D$ 分别为 $\displaystyle n$ 阶和 $\displaystyle m$ 阶可逆矩阵, $\displaystyle B, C$ 分别为 $\displaystyle n \times m$ 和 $\displaystyle m \times n$ 矩阵. 证明:
(1) $\displaystyle \left|\begin{array}{ll}A & B \\ C & D\end{array}\right|=|A|\left|D-C A^{-1} B\right|$ .
(2) $\displaystyle r\left(A-B D^{-1} C\right)-r\left(D-C A^{-1} B\right)=n-m$ .
[例 1.2.11 (南京师范大学, 2020)](https://mp.weixin.qq.com/s?__biz=MzU3OTExMzI1Mg==&mid=2247542182&idx=2&sn=a085010357d788e999dc21cf3ab63c45&chksm=fd697dedca1ef4fbc0f5837d438047e43078d6a4f4635e5baede0aeeebbbfa16f4f1db07ed0d)
例 1.2.12 (中国科学技术大学, 2021) 已知 $\displaystyle \left(\begin{array}{ll}A & B \\ C & D\end{array}\right)$ 为 $\displaystyle n$ 阶实方阵, 且 $\displaystyle B D=D B$ .证明:
$\displaystyle \tiny\boxed{@跟锦数学微信公众号}$
$$ \begin{aligned} \left|\begin{array}{ll}
A & B \\
C & D
\end{array}\right|=|D A-B C|.\tiny\boxed{\begin{array}{c}\mbox{跟锦数学微信公众号}\\\mbox{www.zhangzujin.cn}\end{array}}\end{aligned} $$
[例 1.2.12 (中国科学技术大学, 2021)](https://mp.weixin.qq.com/s?__biz=MzU3OTExMzI1Mg==&mid=2247542202&idx=7&sn=5930600894ef0af4fd5a921fb0a545b8&chksm=fd697df1ca1ef4e731ecc06bb5d72d1f6d86141a3f4ae24eb5f86648c8ae365d529a64ac3c15)
例 1.2.13 (北京邮电大学, 2018) 设 $\displaystyle A$ 可逆, $\displaystyle \alpha, \beta$ 为 $\displaystyle n$ 维列向量, 则
$\displaystyle \tiny\boxed{@跟锦数学微信公众号}$
$$ \begin{aligned} \left|A+\alpha \beta^{\mathrm{T}}\right|=|A|\left(1+\beta^{\mathrm{T}} A^{-1} \alpha\right).\tiny\boxed{\begin{array}{c}\mbox{跟锦数学微信公众号}\\\mbox{www.zhangzujin.cn}\end{array}}\end{aligned} $$
[例 1.2.13 (北京邮电大学, 2018)](https://mp.weixin.qq.com/s?__biz=MzU3OTExMzI1Mg==&mid=2247542202&idx=6&sn=aff3fb9eaa8e4ed6a8d7081e69d53dfb&chksm=fd697df1ca1ef4e77020de9961e0ca5d1964de5fa5d58c7bd4b0c22b01d4c6998b89cc75d471)
例 1.2.14 (中山大学,2022) 求行列式
$\displaystyle \tiny\boxed{@跟锦数学微信公众号}$
$$ \begin{aligned} \left|\begin{array}{lllll}
2 & 1 & 1 & 1 & 1 \\
1 & \frac{3}{2} & 1 & 1 & 1 \\
1 & 1 & \frac{4}{3} & 1 & 1 \\
1 & 1 & 1 & \frac{5}{4} & 1 \\
1 & 1 & 1 & 1 & \frac{6}{5}
\end{array}\right|.\tiny\boxed{\begin{array}{c}\mbox{跟锦数学微信公众号}\\\mbox{www.zhangzujin.cn}\end{array}}\end{aligned} $$
[例 1.2.14 (中山大学,2022)](https://mp.weixin.qq.com/s?__biz=MzU3OTExMzI1Mg==&mid=2247542202&idx=5&sn=133035b86492192b0292591a08a42d10&chksm=fd697df1ca1ef4e7c152b6067efebfeb15393b7b39a5298a9d424e9c20a84c89d536a1cc0790)
例 1.2.15 (哈尔滨工业大学, 2021) 设矩阵 $\displaystyle A=\left(a_{i j}\right)_{6 \times 6 \text {, 其中 }} a_{i i}=2 i, i \neq j$ 时 $\displaystyle a_{i j}=i$ ,求 $\displaystyle A$ 的行列式的值.
[例 1.2.15 (哈尔滨工业大学, 2021)](https://mp.weixin.qq.com/s?__biz=MzU3OTExMzI1Mg==&mid=2247542202&idx=4&sn=4139626911d16299ea0507385f587ce1&chksm=fd697df1ca1ef4e7477458aa970f5f453284f67a5b3bd0c2c85537072a354a44f4dbaa69f7f1)
例 1.2.16 (汕头大学,2012) 计算行列式
$\displaystyle \tiny\boxed{@跟锦数学微信公众号}$
$$ \begin{aligned} D_n=\left|\begin{array}{cccccc}
\frac{3}{2}&1&1&\cdots&1&1 \\
1 & \frac{4}{3} & 1 & \cdots & 1 & 1 \\
1 & 1 & \frac{5}{4} & \cdots & 1 & 1 \\
\vdots & \vdots & \vdots & & \vdots & \vdots \\
1 & 1 & 1 & \cdots & \frac{n+1}{n} & 1 \\
1 & 1 & 1 & \cdots & 1 & \frac{n+2}{n+1}
\end{array}\right|.\tiny\boxed{\begin{array}{c}\mbox{跟锦数学微信公众号}\\\mbox{www.zhangzujin.cn}\end{array}}\end{aligned} $$
[例 1.2.16 (汕头大学,2012)](https://mp.weixin.qq.com/s?__biz=MzU3OTExMzI1Mg==&mid=2247542202&idx=3&sn=25feab5159222193d07be4ed1025ea70&chksm=fd697df1ca1ef4e7cc25684206d9c6ba40b245dbecf94104482fc0f5d24f6201ca40d59502bd)
例 1.2.17 (沈阳工业大学,2022) 计算 $\displaystyle n$ 阶行列式
$\displaystyle \tiny\boxed{@跟锦数学微信公众号}$
$$ \begin{aligned} \left|\begin{array}{ccccc}
2+1 & 2 & 2 & \cdots & 2 \\
2 & 2+\frac{1}{2} & 2 & \cdots & 2 \\
2 & 2 & 2+\frac{1}{3} & \cdots & 2 \\
\vdots & \vdots & \vdots & & \vdots \\
2 & 2 & 2 & \cdots & 2+\frac{1}{n}
\end{array}\right|.\tiny\boxed{\begin{array}{c}\mbox{跟锦数学微信公众号}\\\mbox{www.zhangzujin.cn}\end{array}}\end{aligned} $$
[例 1.2.17 (沈阳工业大学,2022)](https://mp.weixin.qq.com/s?__biz=MzU3OTExMzI1Mg==&mid=2247542202&idx=2&sn=dd99b46f34bce97592ef0f06416913e7&chksm=fd697df1ca1ef4e7a76334c5d43d151286e8f4183513a1bb7c5624f2169b6f4b9a6c149d33b2)
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