- 线代~~已知三阶矩阵A与三维向量X,使得向量组X,AX,A^2X
- 已知三阶矩阵A与三维向量X,使得向量组X,AX,A^2X无关,且满足
A^3X=3AX-2A^2X.
(1)记P=(X,AX,A^2X),求三阶矩阵B,使A=PBP^(-1)
(2)计算行列式|A+E|
- (1)
ⅰ.
X,AX,A^2X无关==>
P=(X,AX,A^2X)可逆.
ⅱ.
AP=(AX,A^2X,A^3X)=
=(AX,A^2X,3AX-2A^2X).
ⅲ.设
B=
[0,0,0]
[1,0,3]
[0,1,-2]
==>
PB=(AX,A^2X,3AX-2A^2X)=AP
==>A=PBP^(-1)
(2)|A+E|=
=|PBP^(-1)+E|=
=|P(B+E)P^(-1)|=
=|B+E|=-4
其中B+E=
[1,0,0]
[1,1,3]
[0,1,-1]