Question

Solve the following system of homogenous equations:

3x + 2y + 7z = 0, 4x – 3y – 2z = 0, 5x + 9y + 23z = 0

Answer 1

Matrix form `[(3, 2, 7),(4, -3, -2),(4, 9, 23)][(x),(y),(z)] = [(0),(0),(0)]`

AX = B

Agumented matrix

[A|B] = `[(3, 2, 7, |, 0),(4, 3, -2, |, 0),(5, 9, 23, |, 0)]`

`{:("R"_2 -> 3"R"_2 - 4"R"_1),("R"_3 -> 3"R"_3 - 5"R"_1),(->):} [(3, 2, 7, |, 0),(0, -17, -34, |, 0),(0, 17, 34, |, 0)]`

`{:("R"-3 -> "R"_3 + "R"_2),(->):} [(3, 2, 7, |, 0),(0, -17, -34, |, 0),(0, 0, 0, |, 0)]`

ρ(A) = 2

ρ[A|B] = 2

ρ(A) ρ[A|B] = 2 < n

The system is consistent.

It has non trivial solution.

Writing the equivalent equations from echelon form

3x + 2y + 7z = 0  .........(1)

– 17y – 34z = 0  .........(2)

Put z = t

(2) ⇒ – 17y = 34t

y = `(34"t")/(-17)` = – 2t

(1) ⇒ 3x + 2(– 2t) + 7t = 0

3x – 4t + 7t = 0

3x + 3t = 0

3x = – 3t

x = – t

(x, y, z) (– t, -2t, t) ∀ t ∈ R