Consider a system of linear equation:
$x - 2 y + 3 z = - 1$
$x - 3 y + 4 z = 1$ and
$- 2 x + 4 y - 6 z = k$
The value of K for which the system has infinitely many solutions is

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Solution

$[A : B] = ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ \begin{matrix} 1 & - 2 & 3 & ⋮ & - 1 1 & - 3 & 4 & ⋮ & 1 - 2 & 4 & - 6 & ⋮ & K \end{matrix} ⎤ ⎥ ⎥ ⎥ ⎥ ⎦$
$R_{2} \to R_{2} - R_{1}, R_{3} \to R_{3} + 2 R_{1}$
$⎡ ⎢ ⎢ ⎢ ⎢ ⎣ \begin{matrix} 1 & - 2 & 3 & ⋮ & - 1 0 & - 1 & 1 & ⋮ & 2 0 & 0 & 0 & ⋮ & K - 2 \end{matrix} ⎤ ⎥ ⎥ ⎥ ⎥ ⎦$
The system has infinite many solutions
$\Rightarrow r a n k (A) = r a n k (A : B) < n$
$\Rightarrow k - 2 = 0$
$k = 2$

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