Question

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

x – y + 2z = 2, 2x + y + 4z = 7, 4x – y + z = 4

Answer 1

Matrix form

`[(1, -1, 2),(2, 1, 4),(4, -1, 1)][(x),(y),(z)] = [(2),(7),(4)]`

AX = B

Augmented matrix

[A|B] = `[(1, -1, 2, |, 2),(2, 1, 4, |, 7),(4, -1, 1, |, 4)]`

`{:("R"_2 -> - 2"R"_1),("R"_3 -> "R"_3 - 4"R"_1),(->):} [(1, -1, 2, |, 2),(0, 3, 0, |, 3),(0, 3, -7, |, -4)]`

`{:("R"_3 -> "R"_3 - "R"_2),(->):} [(1, -1, 2, |, 2),(0, 3, 0, |, 3),(0, 3, -7, |, -7)]`

ρ(A) = 3

ρ[A|B] = 3

The system is consistent.

ρ(A) ρ[A|B] = 3 = n

It has unique solution.

Writing the equivalent equations from echelon form

x – y + 2z = 2  ........(1)

3y = 3

⇒ y = 1

– 7z = – 7

z = 1

(1) ⇒ x – y + 2z = 2

x – 1 + 2 = 2

x = 1

∴ Solution is x = 1, y = 1, z = 1