Question

求此矩阵的秩

Answer 1

a = 1 时， r(A) = 1; a = -1 时，第 2， 4 行相同， r(A) = 3；
a = 0 时，第 2, 3, 4 行均加到第 1 行，然后 A 初等行变换为
[1 1 1 1]
[1 0 1 1]
[1 1 0 1]
[1 1 1 0]
A 初等行变换为
[1 1 1 1]
[0 -1 0 0]
[0 0 -1 0]
[0 0 0 -1]
r(A) = 4;
|a| ≠ 1 且 a ≠ 0 时，A 初等行变换为

[1 1/a 1/a 1/a]
[1 a 1 1]
[1 1 a^2 1]
[1 1 1 a^3]
初等行变换为
[1 1/a 1/a 1/a]
[0 a-1/a 1-1/a 1-1/a]
[0 1-1/a a^2-1/a 1-1/a]
[0 1-1/a 1-1/a a^3-1/a]
初等行变换为
[1 1/a 1/a 1/a]
[0 1-1/a^2 1/a-1/a^2 1/a-1/a^2]
[0 1-1/a a^2-1/a 1-1/a]
[0 1-1/a 1-1/a a^3-1/a]
.................................................................
r(A) = 4