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प्रश्न
Solve the following system of homogenous equations:
2x + 3y – z = 0, x – y – 2z = 0, 3x + y + 3z = 0
उत्तर
Matrix form `[(2, 3, -1),(1, -1, -2),(3, 1, 3)][(x),(y),(z)] = [(0),(0),(0)]`
AX = B
Augmented matrix
[A|B] = `[(2, 3, -1, |, 0),(1, -1, -2, |, 0),(3, 1, 3, |, 0)]`
`{:("R"_1 ↔ "R"_2),(->):} [(1, -1, -2, |, 0),(2, 3, -1, |, 0),(3, 1, 3, |, 0)]`
`{:("R"_2 -> "R"_2 -> 2"R"_1),("R"_3 -> "R"_3 - 3"R"_1),(->):} [(1, -1, -2, |, 0),(0, 5, 3, |, 0),(0, 4, 9, |, 0)]`
`{:("R"_3 -> 5"R"_3 - 4"R"_2),(->):} [(1, -1, -2, |, 0),(0, 5, 3, |, 0),(0, 0, 33, |, 0)]`
ρ(A) = 3
ρ[A|B] = 3
ρ(A) = ρ[A|B] = 3
The system is consistent.
It has trivial solution.
x = 0, y = 0, z = 0
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