6. 行列式

本节, 我要定义本章的主角, 行列式 (hánglièshì). 注意, 我并不为每一个阵定义行列式; 我只为方阵定义行列式.

定义 6.1 (行列式). 设 $A$ 是 $n$ 级阵 ( $n ⩾ 1$ ). 定义 $A$ 的行列式 $det (A) = ⎩ ⎨ ⎧ [A]_{1, 1}, i = 1 \sum n (- 1)^{i + 1} [A]_{i, 1} det (A (i ∣1)), n = 1; n ⩾ 2.$

$det$ 来自英语 determinant.

我们看 4 个例.

例 6.2. 设 $A = [a]$ 是一个 $1$ 级阵. 根据定义, $det (A) = [A]_{1, 1} = a$ .

例 6.3. 设 $A = [a b c d]$ 是一个 $2$ 级阵. 根据定义, $det (A) = = = = (- 1)^{1 + 1} [A]_{1, 1} det (A (1∣1)) + (- 1)^{2 + 1} [A]_{2, 1} det (A (2∣1)) [A]_{1, 1} det [[A]_{2, 2}] - [A]_{2, 1} det [[A]_{1, 2}] [A]_{1, 1} [A]_{2, 2} - [A]_{2, 1} [A]_{1, 2} a d - b c .$ 此事是重要的; 我们会常用它. 通俗地, 我们可用对角线记 $2$ 级阵的行列式.

例 6.4. 设 $A$ 是一个 $3$ 级阵. 根据定义与上个例的结果, $= = = = = det (A) + (- 1)^{1 + 1} [A]_{1, 1} det (A (1∣1)) + (- 1)^{2 + 1} [A]_{2, 1} det (A (2∣1)) + (- 1)^{3 + 1} [A]_{3, 1} det (A (3∣1)) + [A]_{1, 1} det [[A]_{2, 2} [A]_{3, 2} [A]_{2, 3} [A]_{3, 3}] - [A]_{2, 1} det [[A]_{1, 2} [A]_{3, 2} [A]_{1, 3} [A]_{3, 3}] + [A]_{3, 1} det [[A]_{1, 2} [A]_{2, 2} [A]_{1, 3} [A]_{2, 3}] + [A]_{1, 1} ([A]_{2, 2} [A]_{3, 3} - [A]_{3, 2} [A]_{2, 3}) - [A]_{2, 1} ([A]_{1, 2} [A]_{3, 3} - [A]_{3, 2} [A]_{1, 3}) + [A]_{3, 1} ([A]_{1, 2} [A]_{2, 3} - [A]_{2, 2} [A]_{1, 3}) + [A]_{1, 1} [A]_{2, 2} [A]_{3, 3} - [A]_{1, 1} [A]_{3, 2} [A]_{2, 3} - [A]_{2, 1} [A]_{1, 2} [A]_{3, 3} + [A]_{2, 1} [A]_{3, 2} [A]_{1, 3} + [A]_{3, 1} [A]_{1, 2} [A]_{2, 3} - [A]_{3, 1} [A]_{2, 2} [A]_{1, 3} + [A]_{1, 1} [A]_{2, 2} [A]_{3, 3} + [A]_{2, 1} [A]_{3, 2} [A]_{1, 3} + [A]_{3, 1} [A]_{1, 2} [A]_{2, 3} - [A]_{1, 1} [A]_{3, 2} [A]_{2, 3} - [A]_{2, 1} [A]_{1, 2} [A]_{3, 3} - [A]_{3, 1} [A]_{2, 2} [A]_{1, 3} .$ 数学家 Pierre Frédéric Sarrus 给了我们一个记 $3$ 级阵的行列式的好方法. 我们在阵的下方重写此阵. 由左上至右下的对角线 (实线) 上的数的积的和减由左下至右上的对角线 (虚线) 上的数的积的和即为此阵的行列式.

不过, 注意, 前面的对角线法则无法被推广到 $n$ 级阵 ( $n ⩾ 4$ ) 的行列式.

例 6.5. 设

A

是一个

4

级阵. 根据定义,

det (A) = + (- 1)^{1 + 1} [A]_{1, 1} det (A (1∣1)) + (- 1)^{2 + 1} [A]_{2, 1} det (A (2∣1)) + (- 1)^{3 + 1} [A]_{3, 1} det (A (3∣1)) + (- 1)^{4 + 1} [A]_{4, 1} det (A (4∣1)) .

利用跟上个例类似的方法, 并利用上个例的结果, 可知,

det (A)

等于

+ [A]_{1, 1} [A]_{2, 2} [A]_{3, 3} [A]_{4, 4} + [A]_{1, 1} [A]_{3, 2} [A]_{4, 3} [A]_{2, 4} + [A]_{1, 1} [A]_{4, 2} [A]_{2, 3} [A]_{3, 4} - [A]_{1, 1} [A]_{2, 2} [A]_{4, 3} [A]_{3, 4} - [A]_{1, 1} [A]_{3, 2} [A]_{2, 3} [A]_{4, 4} - [A]_{1, 1} [A]_{4, 2} [A]_{3, 3} [A]_{2, 4} - [A]_{2, 1} [A]_{1, 2} [A]_{3, 3} [A]_{4, 4} - [A]_{2, 1} [A]_{3, 2} [A]_{4, 3} [A]_{1, 4} - [A]_{2, 1} [A]_{4, 2} [A]_{1, 3} [A]_{3, 4} + [A]_{2, 1} [A]_{1, 2} [A]_{4, 3} [A]_{3, 4} + [A]_{2, 1} [A]_{3, 2} [A]_{1, 3} [A]_{4, 4} + [A]_{2, 1} [A]_{4, 2} [A]_{3, 3} [A]_{1, 4} + [A]_{3, 1} [A]_{1, 2} [A]_{2, 3} [A]_{4, 4} + [A]_{3, 1} [A]_{2, 2} [A]_{4, 3} [A]_{1, 4} + [A]_{3, 1} [A]_{4, 2} [A]_{1, 3} [A]_{2, 4} - [A]_{3, 1} [A]_{1, 2} [A]_{4, 3} [A]_{2, 4} - [A]_{3, 1} [A]_{2, 2} [A]_{1, 3} [A]_{4, 4} - [A]_{3, 1} [A]_{4, 2} [A]_{2, 3} [A]_{1, 4} - [A]_{4, 1} [A]_{1, 2} [A]_{2, 3} [A]_{3, 4} - [A]_{4, 1} [A]_{2, 2} [A]_{3, 3} [A]_{1, 4} - [A]_{4, 1} [A]_{3, 2} [A]_{1, 3} [A]_{2, 4} + [A]_{4, 1} [A]_{1, 2} [A]_{3, 3} [A]_{2, 4} + [A]_{4, 1} [A]_{2, 2} [A]_{1, 3} [A]_{3, 4} \underline{+ [A]_{4, 1} [A]_{3, 2} [A]_{2, 3} [A]_{1, 4}} .

根据对角线法则, 我们似乎应减由左下至右上的对角线上的数的积. 可是, 由上式,

[A]_{4, 1} [A]_{3, 2} [A]_{2, 3} [A]_{1, 4}

前的符号是

+

, 而不是

-

.

我不要求您记如此具体的公式, 因为 $n$ 级阵 ( $n ⩾ 4$ ) 的行列式的具体的公式不如 $1$ 级阵, $2$ 级阵, $3$ 级阵的行列式的具体的公式实用.

名字空间

视图

6. 行列式