Number of solutions for the system of linear equations, will be,

x + y + z =6

x + 2y + 3z = 14

x + 4y + 2z =30

No solution

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Infinite solution

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Trivial solution only

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Unique Non trivial solutions

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Solution

The correct option is B
Infinite solution

$H e r e A X = B ⎡ ⎢ ⎣ \begin{matrix} 1 & 1 & 1 1 & 2 & 3 1 & 4 & 7 \end{matrix} ⎤ ⎥ ⎦ ⎡ ⎢ ⎣ \begin{matrix} x y z \end{matrix} ⎤ ⎥ ⎦ = ⎡ ⎢ ⎣ \begin{matrix} 61430 \end{matrix} ⎤ ⎥ ⎦ L e t^{'} s c a l c u l a t e | A | | A | = 1 (14 - 12) - 1 (7 - 3) + 1 (4 - 2) = 2 - 4 + 2 = 0 ∴ The given system of equations either has no solution or infinite number of solution C o f a c t o r m a t r i c x o f A w i l l b e = ⎡ ⎢ ⎣ \begin{matrix} 2 & - 4 & 2 - 3 & 6 & - 3 1 & - 2 & 1 \end{matrix} ⎤ ⎥ ⎦ A d j o i n t o f A w i l l b e = ⎡ ⎢ ⎣ \begin{matrix} 2 & - 3 & 1 - 4 & 6 & - 2 2 & - 3 & 1 \end{matrix} ⎤ ⎥ ⎦ ∴ a d j A . B = ⎡ ⎢ ⎣ \begin{matrix} 2 & - 3 & 1 - 4 & 6 & - 2 2 & - 3 & 1 \end{matrix} ⎤ ⎥ ⎦ . ⎡ ⎢ ⎣ \begin{matrix} 61430 \end{matrix} ⎤ ⎥ ⎦ = ⎡ ⎢ ⎣ \begin{matrix} 000 \end{matrix} ⎤ ⎥ ⎦ = 0. We know that if | A | = 0 a n d a d j A . B = 0 t h e n t h e r e e x i s t i n f i n i t e s o l u t i o n s .$

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