例 1.3

计算 $n$ 阶行列式 $(a_{i} \neq = 0)$ :

∣ A ∣ = x_{1} - a_{1} x_{1} x_{1} ⋮ x_{1} x_{2} x_{2} - a_{2} x_{2} ⋮ x_{2} x_{3} x_{3} x_{3} - a_{3} ⋮ x_{3} \dots \dots \dots \dots x_{n} x_{n} x_{n} ⋮ x_{n} - a_{n}

解

第一行乘以 -1 依次加到其余各行上去, 得:

∣ A ∣ = x_{1} - a_{1} a_{1} a_{1} ⋮ a_{1} x_{2} - a_{2} 0 ⋮ 0 x_{3} 0 - a_{3} ⋮ 0 \dots \dots \dots \dots x_{n} 00 ⋮ - a_{n} = a_{1} a_{2} \dots a_{n} \frac{x _{1}}{a _{1}} - 1 11 ⋮ 1 \frac{x _{2}}{a _{2}} - 1 0 ⋮ 0 \frac{x _{3}}{a _{3}} 0 - 1 ⋮ 0 \dots \dots \dots \dots \frac{x _{n}}{a _{n}} 00 ⋮ - 1 .

将后面的列都加到第一列上, 得:

∣ A ∣ = a_{1} a_{2} \dots a_{n} i = 1 \sum n \frac{x _{i}}{a _{i}} - 1 00 ⋮ 0 \frac{x _{2}}{a _{2}} - 1 0 ⋮ 0 \frac{x _{3}}{a _{3}} 0 - 1 ⋮ 0 \dots \dots \dots \dots \frac{x _{n}}{a _{n}} 00 ⋮ - 1

= (- 1)^{n - 1} a_{1} a_{2} \dots a_{n} (i = 1 \sum n \frac{x _{i}}{a _{i}} - 1) . □