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प्रश्न
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
उत्तर
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.
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