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Question
Investigate the values of λ and µ the system of linear equations 2x + 3y + 5z = 9, 7x + 3y – 5z = 8, 2x + 3y + λz = µ, have a unique solution
Solution
Martix form `[(2, 3, 5),(7, 3, -5),(2, 3, lambda)][(x),(y),(z)] = [(0),(8), (mu)]`
AX = B
[A|B] = `[(2, 3, 5, |, 9),(7, 3, -5, |, 8),(2, 3, lambda, |, mu)]`
`{:("R"_2 -> 2"R"_2 - 7"R"_1),("R"_3 -> "R"_3 - "R"_1),(->):} [(2, 3, 5, |, 9),(0, -15, -45, |, -47),(0, 0, lambda - 5, |, mu - 90)]`
Case:
If λ = 5
µ ≠ 9
ρ(A) = 3
ρ(A|B) = 3
ρ(A) = ρ(A|B) = 3 = n
The system is consistent.
It has unique solution.
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