The value of k for which the set of equations`x + ky + 3z=0, 3x + ky – 2z=0, 2x + 3y - 4z=0` has a non-trivial solution over the set of rationals is

Options

A. 15
B. 16
C. `31/2`
D. `33/2`

Answer

To find the value of \( k \) for which the set of equations

\[ \begin{align*} 1. & \quad x + ky + 3z = 0 \\ 2. & \quad 3x + ky - 2z = 0 \\ 3. & \quad 2x + 3y - 4z = 0 \end{align*} \]

has a non-trivial solution, we need to set up the corresponding coefficient matrix and find its determinant. The equations can be expressed in matrix form as follows:

\[ \begin{bmatrix} 1 & k & 3 \\ 3 & k & -2 \\ 2 & 3 & -4 \end{bmatrix} \]

The condition for a non-trivial solution is that the determinant of this matrix must be zero. We will calculate the determinant and set it equal to zero.

### Step 1: Calculate the Determinant

The determinant of a 3x3 matrix

\[ \begin{bmatrix} a & b & c \\ d & e & f \\ g & h & i \end{bmatrix} \]

is given by:

\[ \text{det} = a(ei - fh) - b(di - fg) + c(dh - eg) \]

For our matrix, we have:

\[ \begin{bmatrix} 1 & k & 3 \\ 3 & k & -2 \\ 2 & 3 & -4 \end{bmatrix} \]

Thus, we can substitute \( a = 1, b = k, c = 3, d = 3, e = k, f = -2, g = 2, h = 3, i = -4 \).

Calculating the determinant:

\[ \text{det} = 1 \cdot (k \cdot (-4) - (-2) \cdot 3) - k \cdot (3 \cdot (-4) - (-2) \cdot 2) + 3 \cdot (3 \cdot 3 - k \cdot 2) \]

### Step 2: Simplify the Determinant Expression

Calculating each term:

1. First term:
\[ 1 \cdot (-4k + 6) = -4k + 6 \]

2. Second term:
\[ -k \cdot (-12 + 4) = -k \cdot (-8) = 8k \]

3. Third term:
\[ 3 \cdot (9 - 2k) = 27 - 6k \]

Combining these results, we have:

\[ \text{det} = (-4k + 6) + 8k + (27 - 6k) \]

### Step 3: Combine Like Terms

Now, combine the terms:

\[ \text{det} = -4k + 8k - 6k + 6 + 27 = (-4k + 8k - 6k) + (6 + 27) = -2k + 33 \]

### Step 4: Set the Determinant to Zero

For a non-trivial solution, we set the determinant equal to zero:

\[ -2k + 33 = 0 \]

### Step 5: Solve for \( k \)

Solving for \( k \):

\[ -2k = -33 \implies k = \frac{33}{2} \]

Thus, the value of \( k \) for which the set of equations has a non-trivial solution is

\[ \boxed{\frac{33}{2}} \]