Question

线性代数讨论-2X1+X2+X3=-2 X1-2X2+X3=K X1+X2-2X3=K^2的解的情况，有唯一解，无解和无穷多解（通解）

Answer 1

A =
[ -2, 1, 1, -2]
[ 1, -2, 1, k]
[ 1, 1, -2, k^2]

交换第1,3行
A =
[ 1, 1, -2, k^2]
[ 1, -2, 1, k]
[ -2, 1, 1, -2]

r2-r1;r3+2r1
A =
[ 1, 1, -2, k^2]
[ 0, -3, 3, k - k^2]
[ 0, 3, -3, 2*k^2 - 2]

r3+r2
A =
[ 1, 1, -2, k^2]
[ 0, -3, 3, k - k^2]
[ 0, 0, 0, k^2 + k - 2]

所以k无论取何值，都不会有唯一解
当k=1或k=-2时，有无穷多解
当k≠1或k≠-2时，无解