The system of equations `ax + 4y + z = 0,bx + 3y + z = 0, cx + 2y + z = 0` has non-trivial solution if `a, b, c` are in

Options

A. (a)AP
B. (b)GP
C. (c)HP
D. (d)None of these

Answer

To determine the conditions under which the system of equations

1. \( ax + 4y + z = 0 \)
2. \( bx + 3y + z = 0 \)
3. \( cx + 2y + z = 0 \)

has a non-trivial solution, we can use the concept of determinants. A system of linear equations has a non-trivial solution if the determinant of the coefficients is equal to zero.

### Step-by-step Solution:

1. **Write the system in matrix form**:
The coefficients of the variables \(x\), \(y\), and \(z\) can be represented in a matrix form as follows:
\[ \begin{bmatrix} a & 4 & 1 \\ b & 3 & 1 \\ c & 2 & 1 \end{bmatrix} \]

2. **Set up the determinant**:
We need to calculate the determinant of the coefficient matrix:
\[ D = \begin{vmatrix} a & 4 & 1 \\ b & 3 & 1 \\ c & 2 & 1 \end{vmatrix} \]

3. **Calculate the determinant**:
Using the formula for the determinant of a 3x3 matrix:
\[ D = a \begin{vmatrix} 3 & 1 \\ 2 & 1 \end{vmatrix} - 4 \begin{vmatrix} b & 1 \\ c & 1 \end{vmatrix} + 1 \begin{vmatrix} b & 3 \\ c & 2 \end{vmatrix} \]

Now we calculate each of the 2x2 determinants:
- \( \begin{vmatrix} 3 & 1 \\ 2 & 1 \end{vmatrix} = (3 \cdot 1) - (1 \cdot 2) = 3 - 2 = 1 \)
- \( \begin{vmatrix} b & 1 \\ c & 1 \end{vmatrix} = (b \cdot 1) - (1 \cdot c) = b - c \)
- \( \begin{vmatrix} b & 3 \\ c & 2 \end{vmatrix} = (b \cdot 2) - (3 \cdot c) = 2b - 3c \)

Substituting these back into the determinant:
\[ D = a(1) - 4(b - c) + (2b - 3c) \]
Simplifying this gives:
\[ D = a - 4b + 4c + 2b - 3c = a - 2b + c \]

4. **Set the determinant to zero for non-trivial solutions**:
For the system to have a non-trivial solution, we set the determinant equal to zero:
\[ a - 2b + c = 0 \]

5. **Rearranging the equation**:
Rearranging gives:
\[ a + c = 2b \]

6. **Conclusion**:
This implies that \(a\), \(b\), and \(c\) are in Arithmetic Progression (AP).

### Final Answer:
The system of equations has a non-trivial solution if \(a\), \(b\), and \(c\) are in Arithmetic Progression (AP).