Discuss for all values of k the system of equations.
$x = (k + 4) y + (4 k + 2) z = 0$
$2 x + 3 k y + (3 k + 4) z = 0 o r A X = O$
$x + 2 (k + 1) y + (3 k + 4) z = 0$

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Solution

The given equations can be written as

$⎡ ⎢ ⎢ ⎢ ⎣ \begin{matrix} 1 & k + 4 & 4 k + 2 2 & 3 k & 3 k + 4 1 & 2 k + 2 & 3 k + 4 1 & 2 k + 2 & 3 k + 4 \end{matrix} ⎤ ⎥ ⎥ ⎥ ⎦ ⎡ ⎢ ⎣ \begin{matrix} x y z \end{matrix} ⎤ ⎥ ⎦$ = O

$| A | = ∣ ∣ ∣ ∣ \begin{matrix} 1 & k + 4 & 4 k + 2 2 & 3 k & 3 k + 4 1 & 2 k + 2 & 3 k + 4 \end{matrix} ∣ ∣ ∣ ∣$

Apply $R_{2} - 2 R_{1}, R_{3} - R_{1}$

$∴ | A | = ∣ ∣ ∣ ∣ \begin{matrix} 1 & k + 4 & 4 k + 2 0 & k - 8 & - 5 k 0 & k - 2 & - k + 2 \end{matrix} ∣ ∣ ∣ ∣$ = $(k - 8) (- k + 2) + 5 k (k - 2)$

or $- k^{2} + 2 k + 8 k - 16 + 5 k^{2} - 10 k = 4 k^{2} - 16$

Now $| A | = 0$ when $k^{2} = 4$

$∴ k = + 2 - 2$

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