Question

Test for consistency and if possible, solve the following systems of equations by rank method:

2x – y + z = 2, 6x – 3y + 3z = 6, 4x – 2y + 2z = 4

Answer 1

Matrix form `[(2, -1, 1),(6, -3, 3),(4, -2, 2)][(x),(y),(z)] = [(2),(6),(4)]`

AX = B

Augmented martix

[A|B] = `[(2, -1, 1, |, 2),(6, -3, 3, |, 6),(4, -2, 2, |, 4)]`

`{:("R"_2 -> "R"_2/3),("R"_3 -> "R"_3/2),(->):} [(2, -1, 1, |, 2),(2, -1, 1, |, 2),(2, -1, 1, |, 2)]`

`{:("R"_2 -> "R"_2 - "R"_1),("R"_3 -> "R"_3 - "R"_1),(->):} [(2, -1, 1, |, 2),(0, 0, 0, |, 0),(0, 0, 0, |, 0)]`

ρ(A) = 1

ρ[A|B] = 1

ρ(A) = ρ[A|B] = 1 < n.

∴ The system reduces into a single equation.

∴ It is consistent and has infinitely many solutions.

Writing the equivalent equations from echelon form

2x – y + z = 2

Put y = s, z = t

2x – s + t = 2

2x = 2 + s – t

x = `(2 + "s" - "t")/2`

(x, y, z) = `((2 - "s"- "t")/2, "s", "t")` ∀ s, t ∈ R