Consider the following system of equations

$2 x_{1} + x_{2} + x_{3} = 0$
$x_{2} - x_{3} = 0$
$x_{1} + x_{2} = 0$

This system has

a unique solution

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no solution

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infinite number of solutions

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five solutions

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Solution

The correct option is C infinite number of solutions
Consider A = $⎡ ⎢ ⎣ \begin{matrix} 2 & 1 & 1 0 & 1 & - 1 1 & 1 & 0 \end{matrix} ⎤ ⎥ ⎦$
$| A | = 2 (0 + 1) + (- 1 - 1) = 0$
$\Rightarrow$ The homogeneous system of equations has infinite solutions.

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Q. The system of equations

(1 + λ) x_{1} + x_{2} + x_{3} = 1

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Consider the system of equations
$x_{1} + 2 x_{2} + 3 x_{3} = 1$
$x_{2} + 2 x_{3} + 3 x_{1} = 2$
$x_{3} + 2 x_{1} + 3 x_{2} = 3$
then

A. $x_{1} = \frac{2}{3}$

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C. $x_{3} = - \frac{1}{3}$

D. $x_{1} - x_{2} = \frac{1}{3}$

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Q. If the system of equations

λ x_{1} + x_{2} + x_{3} = 1, x_{1} + λ x_{2} + x_{3} = 1, x_{1} + x_{2} + λ x_{3} = 1

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Consider the following system of equations 2x1+x2+x3=0 x2−x3=0 x1+x2=0 This system has

The correct option is C infinite number of solutionsConsider A = ⎡⎢⎣21101−1110⎤⎥⎦ |A|=2(0+1)+(−1−1)=0 ⇒ The homogeneous system of equations has infinite solutions.

Consider the following system of equations

$2 x_{1} + x_{2} + x_{3} = 0$
$x_{2} - x_{3} = 0$
$x_{1} + x_{2} = 0$

This system has

The correct option is C infinite number of solutions
Consider A = $⎡ ⎢ ⎣ \begin{matrix} 2 & 1 & 1 0 & 1 & - 1 1 & 1 & 0 \end{matrix} ⎤ ⎥ ⎦$
$| A | = 2 (0 + 1) + (- 1 - 1) = 0$
$\Rightarrow$ The homogeneous system of equations has infinite solutions.