设线性方程组
{x1+x2+kx3=-1
{-x1+kx2+x3=k2
{x1-x2+2x3=-4
k取何值时方程组无解?有解?有解的情况求出其全部解．
对增广矩阵作初等行变换，得 (1 1 k ┆ -1 -1 k 1 ┆ k 1 1 2 ┆ -4) → (1 1 k ┆ -1 0 k+1 k+1┆ k2-1 0 -2 2-k ┆ -3) 当k=≠-1时，继续作初等行变换，得 1 1 k ┆ -1 0 1 1 ┆ k-1 0 -2 2-k ┆ -3 → 1 0 k-1 ┆ -k 0 1 1 ┆ k-1 0 0 4-k ┆ 2k-5 如果k=4，则出现矛盾方程，无解；如果k≠4，继续作初等行变换， 得 1 0 0 ┆ (-k2+3k-5)/(4-k) 0 1 0 ┆ (-k2+3k-9)/(4-k) 0 0 1 ┆ (2k-5)/(4-k) 方程组有唯一解，解为． x= {x1=(-k2+3k-5)/(4-k) {x2=(-k2+3k-9)/(4-k) (x3=(2k-5)/(4-k) 当k=-1时，增广矩阵经过初等变换后为 1 1 -1 ┆ -1 0 0 0 ┆ 0 0 -2 3 ┆-3 → 1 0 1/2 ┆ -(5/2) 0 1 -(3/2) ┆ 3/2 0 0 0 ┆ 0 同解方程组为 {x1 +(1/2)x3=1(5/2) { x2-(3/2)x3=3/2 x3为自由未知量，方程组有无穷多解，全部解为 {x1=-(5/2)-(1/2)k {x2=3/2+(3/2) (k为任意实数) {x3=k
