The given system of equation are ‌(λ−1)x+(λ−4)y+λz=5 ‌λx+(λ−1)y+(λ−4)z=7 ‌(λ+1)x+(λ+2)y−(λ+2)z=9 has infinite many solutions, then D=D1=D2=D3=0 ∴|
λ−1
λ−4
λ
λ
λ−1
λ−4
λ+1
λ+2
−(λ+2)
|=0 or, |
−1
−3
4
−1
−3
2λ−2
λ+1
λ+2
−λ−2
|=0 or, |
−1
−3
4
0
0
2λ−6
λ+1
λ+2
−λ−2
|=0
‌∴‌‌(6−2λ)(2λ+1)=0 ‌∴‌‌λ=3‌ or ‌−‌
1
2
But λ=3 satisfies all other conditions. ∴‌‌λ2+λ=12