Question

Show that the following system of equations have unique solutions: x + y + z = 3, x + 2y + 3z = 4, x + 4y + 9z = 6 by rank method

Answer 1

x + y + z = 3

x + 2y + 3z = 4

x + 4y + 9z = 6

The matrix equation corresponding to the given system is

`[(1, 1, 1),(1, 2, 3),(1, 4, 9)][(x),(y),(z)] = [(3),(4),(6)]`
        A           X     =    B

Augmented Matrix [A, B]	Elementary Transformation
`[(1, 1, 1, 3),(1, 2, 3, 4),(1, 4, 9, 6)]`
`∼ [(1, 1, 1, 3),(0, 1, 2, 1),(0, 3, 8, 3)]`	`{:("R"_2 -> "R"_2 - "R"_1),("R"_3 -> "R"_3 -"R"_1):}`
`∼ [(1, 1, 1, 3),(0, 1, 2, 1),(0, 0, 2, 0)]`	`{:"R"_3 -> "R"_3 - 3"R"_2:}`
p(A) = 3; p(A , B) = 3

The last equivalent matrix is in the echelon form [A, B] has 3 non-zero rows and [A] has 3 non-zero rows.

p([A,B]) = 3; ρ(A) = 3

ρ([A, B]) = ρ(A) = No. of unknowns

∴ The system of equations have unique solution.