1

计算下列行列式:

1.1

$sin x cos x - cos x sin x;$

1.2

$sin x cos x cos x 2 cos x$ ;

1.3

$147258369$ ;

1.4

$12 - 3 - 1 0 - 3 - 2 53$ ;

1.5

$x y y y x y y y x$ .

2

证明下列等式:

2.1

$a c b + x d + y = a c b d + a c x y$

2.2

$010 b e d a f c = a c b d$

2.3

$a_{11} a_{21} a_{31} a_{12} a_{22} a_{32} a_{13} a_{23} a_{33} = a_{11} a_{12} a_{13} a_{21} a_{22} a_{23} a_{31} a_{32} a_{33} = - a_{11} a_{31} a_{21} a_{12} a_{32} a_{22} a_{13} a_{33} a_{23}$

3

借助行列式的知识解下列线性方程组:

3.1

$⎩ ⎨ ⎧ 3 x_{1} + 2 x_{2} - 4 x_{3} x_{1} + 2 x_{2} - x_{3} 2 x_{1} - x_{2} + x_{3} = - 6, = 3, = 17 .$

3.2

$⎩ ⎨ ⎧ 2 x_{1} - x_{2} 5 x_{1} - x_{3} - 2 x_{2} + x_{3} = 0, = 0, = 3.$

4

求相应的 $i, j$ 值,使:

4.1

$17 i 52 j 6$ 为偶排列;

4.2

$246 i 891 j 7$ 为奇排列.

5

如果排列 $i_{1} i_{2} \dots i_{n}$ 的逆序数为 $m$ ,求排列 $i_{n} i_{n - 1} \dots i_{2} i_{1}$ 的逆序数.

6

计算下列各排列的逆序数并判断排列的奇偶性:

6.1

26538417 ;

6.2

$n (n - 1) \dots 21$

6.3

$2 n (2 n - 2) \dots 2 (2 n - 1) (2 n - 3) \dots 1$

7

写出 5 阶行列式 $∣ a_{ij} ∣$ 中含有因子 $a_{12} a_{35} a_{41}$ 的项.

8

在多项式 $f (x) = x 102 74 x 1 3 x - 1 2 - 1 053$ 中,求二次项 $x^{2}$ 的系数.

9

证明: 如果 $n$ 阶行列式 $D$ 含有多于 $n^{2} - n$ 个元素为零,则 $D = 0$ .

10

用行列式的定义计算下列行列式:

10.1

$0 b 00 a 0 c 0 000 e a 0 d 0$

10.2

$000 g 000 f 000 e a b c d$

10.3

$a 00 b 0 a b 0 0 b a 0 b 00 a$

10.4

$a_{11} 0 ⋮ 00 00 ⋮ 0 a_{n 2} \dots \dots \dots \dots 00 ⋮ a_{n - 1, n - 1} a_{n, n - 1} 0 a_{2 n} ⋮ a_{n - 1, n} a_{nn}$

10.5

$a_{1} 10 ⋮ 00 a_{2} 01 ⋮ 00 \dots \dots \dots \dots \dots a_{n - 2} 00 ⋮ 10 a_{n - 1} 00 ⋮ 01 a_{n} 00 ⋮ 00$

11

利用行列式的性质计算下列行列式:

11.1

$10132513261201525113$

11.2

$13002614312374559$

11.3

$1234234134124123$

11.4

$1^{3} 4^{3} 3^{3} 2^{3} 2^{3} 1^{3} 4^{3} 3^{3} 3^{3} 2^{3} 1^{3} 4^{3} 4^{3} 3^{3} 2^{3} 1^{3}$

11.5

$x x ⋮ x a x x ⋮ a x \dots \dots \dots \dots x a ⋮ x x a x ⋮ x x_{n}$

11.6

$x 0 ⋮ 0 a x 0 ⋮ a 0 \dots \dots \dots \dots x a ⋮ 00 a x ⋮ x x_{n}$

12

证明下列等式:

12.1

$21 - 1 l g 4 cos^{2} α - 2 2 l g 5 sin^{2} α \frac{1}{2 + 1} = 0$

12.2

$x_{1} + y_{1} y_{1} + z_{1} z_{1} + x_{1} x_{2} + y_{2} y_{2} + z_{2} z_{2} + x_{2} x_{3} + y_{3} y_{3} + z_{3} z_{3} + x_{3} = 2 x_{1} x_{2} x_{3} y_{1} y_{2} y_{3} z_{1} z_{2} z_{3}$

12.3

$a 11 ⋮ 1 2 a + 1 2 ⋮ 2 33 a + 2 ⋮ 3 \dots \dots \dots \dots n n n ⋮ a + n - 1 = [a + \frac{( n - 1 ) ( n + 2 )}{2}] (a - 1)^{n - 1}$

12.4

$a + b 10 ⋮ 00 ab a + b 1 ⋮ 00 0 ab a + b ⋮ 00 \dots \dots \dots \dots \dots 000 ⋮ a + b 1 000 ⋮ ab a + b_{n} = i = 0 \sum n a^{i} b^{n - i}$

13

设有 $n$ 阶行列式 $D = det (a_{ij})$ ,若其元素满足 $a_{ji} = - a_{ij}$ ,则称为反称行列式. 试证明:

反称行列式主对角线上的元素全为零;
奇数阶反称行列式的值必为零.

14

计算下列行列式:

14.1

$70 - 3 - 2 4920611 10 - 1 - 3 1051$

14.2

$a_{11} a_{21} 00 a_{12} a_{22} 00 a_{13} a_{23} a_{33} a_{43} a_{14} a_{24} a_{34} a_{44}$

14.3

$x y 0 ⋮ 00 0 x y ⋮ 00 00 x ⋮ 00 \dots \dots \dots \dots \dots 000 ⋮ x y y 00 ⋮ 0 x_{n}$

14.4

$x y 0 ⋮ 00 z x y ⋮ 00 0 z x ⋮ 00 \dots \dots \dots \dots \dots 000 ⋮ x y 000 ⋮ z x_{n}$

15

计算行列式: $1 - 1 1 - 1 1 124816 1 - 3 9 - 27 81 1 \frac{2}{2} \frac{1}{2} \frac{2}{4} \frac{1}{4} 11111 .$

16

证明: $a^{2} b^{2} c^{2} d^{2} (a + 1)^{2} (b + 1)^{2} (c + 1)^{2} (d + 1)^{2} (a + 2)^{2} (b + 2)^{2} (c + 2)^{2} (d + 2)^{2} (a + 3)^{2} (b + 3)^{2} (c + 3)^{2} (d + 3)^{2} = 0.$

17

计算行列式: $x^{n} (x + 1)^{n} (x + 2)^{n} ⋮ (x + n - 1)^{n} (x + n)^{n} x^{n - 1} (x + 1)^{n - 1} (x + 2)^{n - 1} ⋮ (x + n - 1)^{n - 1} (x + n)^{n - 1} \dots \dots \dots \dots \dots x^{2} (x + 1)^{2} (x + 2)^{2} ⋮ (x + n - 1)^{2} (x + n)^{2} x x + 1 x + 2 ⋮ x + n - 1 x + n 111 ⋮ 11 .$

18

解下列线性方程组:

18.1

$⎩ ⎨ ⎧ x_{1} + 3 x_{2} + 2 x_{3} = 7, 2 x_{1} + x_{2} + 4 x_{3} = 6, 3 x_{1} + 2 x_{2} + x_{3} = 4;$

18.2

$⎩ ⎨ ⎧ 3 x_{1} + x_{2} + 2 x_{3} 5 x_{2} + x_{3} + 4 x_{4} 4 x_{1} + 2 x_{3} + x_{4} 2 x_{1} + x_{2} + x_{4} = 0, = 1, = 0, = 1;$

18.3

⎩ ⎨ ⎧ x_{1} + 6 x_{2} + 2 x_{3} - x_{4} + x_{5} x_{1} - 2 x_{2} - x_{3} 2 x_{1} - 3 x_{3} + x_{4} + 3 x_{5} x_{3} + 2 x_{5} x_{2} + x_{3} - 3 x_{4} = 0, = 0, = 0, = 0, = 0.

19

试用数学归纳法证明:

= a_{11} ⋮ a_{n 1} c_{11} ⋮ c_{m 1} \dots \dots \dots \dots a_{1 n} ⋮ a_{nn} c_{1 n} ⋮ c_{mn} 0 ⋮ 0 b_{11} ⋮ b_{m 1} \dots \dots \dots \dots 0 ⋮ 0 b_{1 m} ⋮ b_{mm} a_{11} ⋮ a_{n 1} \dots \dots a_{1 n} ⋮ a_{nn} \cdot b_{11} ⋮ b_{m 1} \dots \dots b_{1 m} ⋮ b_{mm} .

20

如果齐次线性方程组 $⎩ ⎨ ⎧ λ x_{1} + x_{2} + x_{3} x_{1} + λ x_{2} + 2 x_{3} λ^{2} x_{1} + x_{2} + λ x_{3} = 0, = 0, = 0$ 有非零解,求 $λ$ 的值.

21

设 $a_{1}, a_{2}, \dots, a_{n}$ 是互不相同的实数, $b_{1}, b_{2}, \dots, b_{n}$ 是任意实数, 用克拉默法则证明: 存在唯一的实系数多项式 $f (x) = c_{n - 1} x^{n - 1} + c_{n - 2} x^{n - 2} + \dots + c_{1} x + c_{0}$ 使得 $f (a_{i}) = b_{i} (i = 1, 2, \dots, n) .$

Youliang Zhong

习题一

1

1.1

1.2

1.3

1.4

1.5

2

2.1

2.2

2.3

3

3.1

3.2

4

4.1

4.2

5

6

6.1

6.2

6.3

7

8

9

10