4. When we ask what value a takes, the following vector groups are linearly correlated? A1 = (a, 1,1), A2 = (1, a, - 1) t, A3 = (1, - 1, a) t

4. When we ask what value a takes, the following vector groups are linearly correlated? A1 = (a, 1,1), A2 = (1, a, - 1) t, A3 = (1, - 1, a) t

det(a1,a2,a3) =0
a 1 1
1 a -1
1 -1 a
=> a^3-1-1-(a+a+a)=0
a^3-3a-2 =0
(a+1)(a^2-a-2)=0
(a+1)^2(a-2)=0
a=-1 or 2

Linear algebra, let A1, A2, A3 be three-dimensional column vectors, and | A1, A2, A3 | = 1, then | A3, A2, a1-2a2 | = 1

|a3,a2,a1-2a2|
c3+2c2
＝ |a3,a2,a1|
c1c3
＝ - |a1,a2,a3 |
＝ -1.

Let vector group A1, A2, A3 be linearly independent, and prove that vector group a1-a2, A2-A3, a3-a1 are linearly related