设有线性方程组x1+3x2+x3=0,3x1+2x2+3x3=-1,-x1+4x2+mx3=k,问mk何值是唯一解无穷解及无穷时一般解x1+3x2+x3=0,3x1+2x2+3x3=-1,-x1+4x2+mx3=k急~

Question

设有线性方程组x1+3x2+x3=0,3x1+2x2+3x3=-1,-x1+4x2+mx3=k,问mk何值是唯一解无穷解及无穷时一般解
x1+3x2+x3=0,
3x1+2x2+3x3=-1,
-x1+4x2+mx3=k
急~

Answer 1

增广矩阵=
1 3 1 0
3 2 3 -1
-1 4 m k
r2-3r1,r3+r1
1 3 1 0
0 -7 0 -1
0 7 m+1 k
r3+r2
1 3 1 0
0 -7 0 -1
0 0 m+1 k-1
当m≠-1时,方程组有唯一解
当m=-1且k≠1时,方程组无解
当m=-1且k=1时,方程组有无穷多解
此时增广矩阵化为
1 3 1 0
0 -7 0 -1
0 0 0 0
-->
1 0 1 -3/7
0 1 0 1/7
0 0 0 0
方程组的一般为:(-3/7,1/7,0)^T+c(1,0,-1)^T.