We have, x+y+z=3 x+2y+2z=6 x+ay+3z=b Above equations can be written as AX=B Where,
A=[
1
1
1
1
2
2
1
a
3
],X=[
x
y
z
],B=[
3
6
b
]
Now, |A|=|
1
1
1
1
2
2
1
a
3
| =1[6−2a]−1[3−2]+1[a−2] =6−2a−1+a−2=3−a For unique solution |A|≠0⇒3−a≠0⇒a≠3 ∴ Given system have unique solution when a≠3,b can have any value.