Question

用行列式的性质计算下列行列式 2-x 2 -2 2 5-x -4 -2 -4 5-x

Answer 1

第3行加上第2行，得到
D=
2-x
2
-2
2
5-x
-4
0
1-x
1-x
第2列减去第3列
=
2-x
4
-2
2
9-x
-4
0
0
1-x
按第3行展开
=
(1-x)*[8
-(2-x)(9-x)]
=(1-x)(-x^2+11x
-10)
=x^3-11x^2+10x
-x^2+11x
-10
=x^3-12x^2+21x
-10
故行列式的值为
x^3-12x^2+21x
-10

Answer 2

|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)