例 1.5

计算下列 $n$ 阶行列式:

∣ A ∣ = a 0 ⋮ 01 0 a ⋮ 00 \dots \dots \dots \dots 00 ⋮ a 0 10 ⋮ 0 a

解

按第一列展开, 经计算得

∣ A ∣ = a^{n} + (- 1)^{n + 1} 0 a ⋮ 0 00 ⋮ 0 \dots \dots \dots 00 ⋮ a 10 ⋮ 0 = a^{n} - a^{n - 2} .