试确定a为何值时,使得向量组a1(3,a,0),a2(a,1,2),a3(1,-2,1),a4(2,-4,2)的秩为3

题目

试确定a为何值时,使得向量组a1(3,a,0),a2(a,1,2),a3(1,-2,1),a4(2,-4,2)的秩为3

答案

a1(3,a,0),a2(a,1,2),a3(1,-2,1),a4(2,-4,2)
因为a3和a4对应成比例
所以
只要 a1,a2,a3线性无关即可
也即
| 3 a 0
a 1 2
1 -2 1|≠0即可
=| 3 a 0
a-2 5 0
1 -2 1|
=3×5-a(a-2)
=-a²+2a+15≠0
a²-2a-15≠0
(a+3)(a-5)≠0
即
a≠-3且a≠5.