Question

已知3阶矩阵A有3维向量A满足A^3X=3AX-A^2X,且向量组X,AX,A^2X线性无关.（1）记P=(X,AX,A

Answer 1

(1) AP = A(X,AX,A^2X)

= (AX,A^2X,A^3X)

= (AX,A^2X,3AX-A^2X)

= (X,AX,A^2X)B ＝ PB.

其中 B ＝

0 0 0

1 0 3

0 1 -1

(2) 易知 |B| = 0.

由向量组X,AX,A^2X线性无关, 所以P可逆

所以 A = PBP^-1

所以 |A| = |P||B||P^-1| = 0.