Statement 1: If the system of equations x + k | vedclass

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(A) For a system of homogeneous linear equations to have a nontrivial solution,the determinant of the coefficient matrix must be zero.The coefficient

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Statement $1$ is false,Statement $2$ is true.

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(A) For a system of homogeneous linear equations to have a nontrivial solution,the determinant of the coefficient matrix must be zero.The coefficient matrix is $A = \begin{bmatrix} 1 & k & 3 \ 3 & k & -2 \ 2 & 3 & -4 \end{bmatrix}$.Setting the determinant to zero: $\left| \begin{matrix} 1 & k & 3 \ 3 & k & -2 \ 2 & 3 & -4 \end{matrix} ight| = 0$.Expanding along the first row:$1(-4k - (-6)) - k(-12 - (-4)) + 3(9 - 2k) = 0$$1(-4k + 6) - k(-8) + 27 - 6k = 0$$-4k + 6 + 8k + 27 - 6k = 0$$-2k + 33 = 0$$2k = 33 \Rightarrow k = \frac{33}{2}$.Since the calculated value of $k$ is $\frac{33}{2}$ and not $\frac{31}{2}$,Statement $1$ is false.Statement $2$ is a standard theorem in linear algebra regarding homogeneous systems,so it is true.

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