The system of equations are x+3y+5z=αx 5x+y+3z=αy 3x+5y+z=αz ⇒(1−α)x+3y+5z=0 ....(i) 5x+(1−α)y+3z=0....(ii) 3x+5y+(1−α)z=0 ....(iii) For infinite number of solutions, we must have |
1−α
3
5
5
1−α
3
3
5
1−α
|=0
⇒|
9−α
3
5
9−α
1−α
3
9−α
5
1−α
|=0
C1→C1+C2+C3 Taking (9 -α)common from the first column, we get
⇒(9−α)|
1
3
5
1
1−α
3
1
5
1−α
|=0 ⇒(9−α)|
1
3
5
0
−α−2
−2
0
2
−α−4
|=0
⇒(9−α)|
α+2
2
−2
α+4
|=0 ⇒(9−α)(α2+6α+8+4)=0 (α−9)(α2+6α+12)=0 ⇒α=9 is the only real value of a.