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Determinant

Consider the ...

Question

Consider the system of equation
$a x + b y + c z = 2$
$b x + c y + a z = 2$
$c x + a y + b z = 2$
where $a, b, c$ are real numbers such that $a + b + c = 0$
Then the system

has two solutions

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is inconsistent

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has unique solution

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has infinitely many solution

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Solution

The correct option is A is inconsistent
$⎡ ⎢ ⎣ \begin{matrix} a & b & c & | & 2 b & c & a & | & 2 c & a & b & | & 2 \end{matrix} ⎤ ⎥ ⎦$

$R_{1} \to R_{1} + R_{2} + R_{3}$

$= ⎡ ⎢ ⎣ \begin{matrix} a + b + c & b + c + a & c + a + b & | & 6 b & c & a & | & 2 c & a & b & | & 2 \end{matrix} ⎤ ⎥ ⎦$

$= ⎡ ⎢ ⎣ \begin{matrix} 0 & 0 & 0 & | & 6 b & c & a & | & 2 c & a & b & | & 2 \end{matrix} ⎤ ⎥ ⎦$ -------- $(A s a + b + c = 0)$
From the first row, we get the equation $0 x + 0 y + 0 z = 6$
This equation does not have any solution.
Hence the system of equations is inconsistent.
Answer is option (B)

Suggest Corrections

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