Question

For what values of the parameter λ, will the following equations fail to have unique solution: 3x – y + λz = 1, 2x + y + z = 2, x + 2y – λz = – 1

Answer 1

3x – y + λz = 1

2x + y + z = 2

x + 2y – λz = – 1

The matrix equation corresponding to the given system is

`[(3, -1, lambda),(2, 1, 1),(1, 2, -lambda)] [(x),(y),(z)] = [(1),(2),(-1)]`
          A                X    =      B

Augmented Matrix [A, B]	Elementary Transformation
`[(3, -1, lambda, 1),(2, 1, 1, 2),(1, 2, -lambda, -1)]`
`∼[(1, 2, -lambda, -1),(2, 1, 1, 2),(3, -1, lambda, 1)]`	`{:"R"_1 ↔ "R"_3:}`
`∼[(1, 2, -lambda, -1),(0, -3, 1 + 2lambda, 4),(0, -7, 4lambda, 4)]`	`{:("R"_2 -> "R"_2 - 2"R"_1),("R"_3 -> "R"_3 - 3"R"_1):}`
`∼[(1, 2, -lambda, -1),(0, -3, 1 + 2lambda, 4),(0, -1, -2, -4)]`	`{:"R"_3 -> "R"_3 - 2"R"_2:}`
`∼[(1, 2, -lambda, -1),(0, -1, -2, -4),(0, -3, 1 + 2lambda, 4)]`	`{:"R"_2 ↔ "R"_3:}`
`∼[(1, 2, -lambda, -1),(0, -1, -2, -4),(0, 0, 7 + 2lambda, 16)]`	`{:"R"_3 -> "R"_3 - 3"R"_2:}`

If the equations fail to have unique solution.

ρ(A) ≠ ρ(A, B)

ρ(A, B) = 3

ρ(A) ≠ 3

Therefore 7 + 2λ = 0

2λ = – 7 and λ = `(-7)/2`