Question

求这个行列式5-x 2 2 2 6-x 0 2 0 4-x这个式的值为(5-x)(2-x)(8-x),怎么来的,3乘3的

Answer 1

这是求特征值出现的行列式

这个没有好方法

|A-λE| =

5-λ 2 2

2 6-λ 0

2 0 4-λ

= (4-λ)(5-λ)(6-λ)-4(6-λ)-4(4-λ) --直接对角线法则得

= (4-λ)(5-λ)(6-λ)-4(10-2λ) --后两项合并得因子(5-λ)

= (4-λ)(5-λ)(6-λ)-8(5-λ)

= (5-λ)[(4-λ)(6-λ)-8]

= (5-λ)[λ^2-10λ+16] -- 十字相乘法分解

= (5-λ)(2-λ)(8-λ)