Question

求解线性方程组当a为何值时,线性方程组有唯一解,无解,无穷多解x1+2x2+x3=-1,x1+3x2+(a+1)x3=0

Answer 1

x3=1-ax2,x1=-1-2x2-x3=-1-2x2-(1-ax2)=-2+(a-2)x2

-2+(a-2)x2+3x2+(a+1)(1-ax2)=0=(1-a^2)x2+a-1=(a-1)[1-(a+1)x2]

所以a=1时有无穷多的解x1=-2-x2,x3=1-x2

a=-1时会推出a=1矛盾,所以a=1时无解

a≠±1时,x2=1/(a+1),x1=-2+(a-2)/(a+1)=-(a+4)/(a+1)=-1-3/(a+1),x3=1/(a+1)