Simultaneous Linear Equations
Given the linear system:
2 x 1 + x 2 − x 3 + x 4 − 3 x 5 x 1 + 2 x 3 − x 4 + x 5 − 2 x 2 − x 3 + x 4 − x 5 3 x 1 + x 2 − 4 x 3 + 5 x 5 x 1 − x 2 − x 3 − x 4 + x 5 = 7 , = 2 , = − 5 , = 6 , = 3 .
Find the value of ⌊ Ψ ⌉ where
Ψ = 1 0 0 0 ∣ x 1 + x 2 + x 3 + x 4 + x 5 ∣ .
The answer is 1433.
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Employing matrices here, we can write:
⎣ ⎢ ⎢ ⎢ ⎢ ⎡ 2 1 0 3 1 1 0 − 2 1 − 1 − 1 2 − 1 − 4 − 1 1 − 1 1 0 − 1 − 3 1 − 1 5 1 ⎦ ⎥ ⎥ ⎥ ⎥ ⎤ ⋅ ⎣ ⎢ ⎢ ⎢ ⎢ ⎡ x 1 x 2 x 3 x 4 x 5 ⎦ ⎥ ⎥ ⎥ ⎥ ⎤ = ⎣ ⎢ ⎢ ⎢ ⎢ ⎡ 7 2 − 5 6 3 ⎦ ⎥ ⎥ ⎥ ⎥ ⎤
which has the unique solution:
⎣ ⎢ ⎢ ⎢ ⎢ ⎡ x 1 x 2 x 3 x 4 x 5 ⎦ ⎥ ⎥ ⎥ ⎥ ⎤ = ⎣ ⎢ ⎢ ⎢ ⎢ ⎡ 3 2 8 / 1 7 1 3 3 6 / 1 7 1 − 1 6 9 / 1 7 1 − 5 4 6 / 1 7 1 − 1 9 4 / 1 7 1 ⎦ ⎥ ⎥ ⎥ ⎥ ⎤
and Ψ = 1 0 0 0 ∣ 1 7 1 3 2 8 + 3 3 6 − 1 6 9 − 5 4 6 − 1 9 4 ∣ = 1 4 3 2 . 7 4 8 ⇒ ⌈ Ψ ⌋ = 1 4 3 3 .