Question

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

2x + 2y + z = 5, x – y + z = 1, 3x + y + 2z = 4

Answer 1

Matrix form `[(2, 2, 1),(1, -1, 1),(3, 1, 2)][(x),(y),(z)] = [(5),(1),(4)]`

AX = B

Augmented matrix

[A|B] = `[(2, 2, 1, |, 5),(1, -1, 1, |, 1),(3, 1, 2, |, 4)]`

`{:("R"_1 ↔ "R"_2),(->):} [(1, -1, 1, |, 1),(2, 2, 1, |, 5),(3, 1, 2, |, 4)]`

`{:("R"_2 -> "R"_2 - 2"R"_1),("R"_3 -> "R"_3 - 3"R"_1),(->):} [(1, -1, 1, |, 1),(0, 4, -1, |, 3),(0, 4, -1, |, 1)]`

`{:("R"_3 -> "R"_3 - "R"_2),(->):} [(1, -1, 1, |, 1),(0, 4, -1, |, 3),(0, 0, 0, |, -2)]`

ρ(A) = 2

ρ[A|B] = 3

ρ(A) ≠ ρ[A|B] = 2 < n

∴ The system is inconsistent. It has no solution.