Find the value of B for which the system of equations x+y+z=1 x + 2y + 3z =B x + 5y + 9z = B2
is consistent. Also find the solution of the system for each value of ß obtained
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Step-by-step explanation:
The given system can be written as
⎣
⎢
⎢
⎡
1
1
1
1
2
2
1
3
λ
⎦
⎥
⎥
⎤
⎣
⎢
⎢
⎡
x
y
z
⎦
⎥
⎥
⎤
=
⎣
⎢
⎢
⎡
6
10
μ
⎦
⎥
⎥
⎤
AX=B
The augmented matrix.
(A,B)=
⎣
⎢
⎢
⎡
1
1
1
1
2
2
1
3
λ
6
10
μ
⎦
⎥
⎥
⎤
=
⎣
⎢
⎢
⎡
1
1
1
1
2
2
1
3
λ−3
6
4
μ−10
⎦
⎥
⎥
⎤
R
2
→R
2
−R
1
R
3
→R
3
−R
2
Case (i) : λ−3=0 and μ−10
=0
(ie) λ=3,μ
=10
ρ(A)=2,ρ(A,B)=3
∴ρ(A)
=ρ(A,B)
The given system is inconsistent but has no solution.
Case (ii) : λ−3
=0 and μ∈R
(ie) λ
=3
\rho(A) = \rho(A,B) =3$$
The given system is consistent and has unique solution.
Case (iii) : λ=3 and μ=10
ρ(A)=ρ(A,B)=2< number of unknowns.
The given system is consistent but has an infinite number of solutions.
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