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Question
Determine the values of λ for which the following system of equations x + y + 3z = 0; 4x + 3y + λz = 0, 2x + y + 2z = 0 has a unique solution
Solution
Matrix form `[(1, 1, 3),(4, 3, lambda),(2, 1, 2)][(x),(y),(z)] = [(0),(0),(0)]`
AX = B
Augmented matrix
[A|B] = `[(1, 1, 3, |, 0),(4, 3, lambda, |, 0),(2, 1, 2, |, 0)]`
`{:("R"_2 -> "R"_2 - 4"R"_1),("R"_3 -> "R"_3 - 2"R"_2),(->):} [(1, 1, 3, |, 0),(0, -1, lambda - 12, |, 0),(0, -1, -4, |, 0)]`
`{:("R" _2 ↔ "R"_3),(->):} [(1, 1, 3, |, 0),(0, -1, -4, |, 0),(0, -1, lambda - 12, |, 0)]`
`{:("R"_3 -> "R"_3 - "R"_2),(->):} [(1, 1, 3, |, 0),(0, -1, -4, |, 0),(0, 0, lambda - 8, |, 0)]`
Case:
if λ ≠ 8
ρ(A) = 3
ρ(A|B) = 3
ρ(A) = ρ(A|B) = 3 = n
The system is consistent.
It has unique (trivial) solution.
∴ Solution x = 0, y = 0, z = 0
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