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(D) For a system of homogeneous linear equations to have a non-trivial solution,the determinant of the coefficient matrix must be equal to zero.The sy

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$\frac{33}{2}$

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(D) For a system of homogeneous linear equations to have a non-trivial solution,the determinant of the coefficient matrix must be equal to zero.The system is:$1x + ky + 3z = 0$$3x + ky - 2z = 0$$2x + 3y - 4z = 0$The determinant of the coefficient matrix is:$\left|\begin{array}{ccc} 1 & k & 3 \ 3 & k & -2 \ 2 & 3 & -4 \end{array}ight| = 0$Expanding the determinant along the first row:$1((-4)(k) - (3)(-2)) - k((3)(-4) - (2)(-2)) + 3((3)(3) - (2)(k)) = 0$$1(-4k + 6) - k(-12 + 4) + 3(9 - 2k) = 0$$-4k + 6 + 8k + 27 - 6k = 0$$-2k + 33 = 0$$2k = 33$$k = \frac{33}{2}$

For what value of $k$ does the following system of equations possess a non-trivial solution?
$x + ky + 3z = 0$
$3x + ky - 2z = 0$
$2x + 3y - 4z = 0$

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For what value of $k$ does the following system of equations possess a non-trivial solution?$x + ky + 3z = 0$$3x + ky - 2z = 0$$2x + 3y - 4z = 0$