矩阵A是4x4阶的矩阵A=[a111/1a11/11a1/111a],r(A)=3,a=?

Question

矩阵A是4x4阶的矩阵A=[a 1 1 1/1 a 1 1/1 1 a 1/1 1 1 a],r(A)=3,a=?

Answer 1

r1-r2,r2-r3,r3-r4
a-1 1-a 0 0
0 a-1 1-a 0
0 0 a-1 1-a
1 1 1 a
若 a=1 则 R(A) = 1. 故a≠1.
第1,2,3行乘 1/(a-1)
1 -1 0 0
0 1 -1 0
0 0 1 -1
1 1 1 a

r3-r1-2r2-3r3
1 -1 0 0
0 1 -1 0
0 0 1 -1
0 0 0 a+3

由 r(A) = 3 所以 a=-3.
满意请采纳 ^_^