If 3x + 2y + z = 0, x + 4y + z = 0, 2x + y + 4z = 0 be a system of equations, thena)it is inconsistentb)it has only the trivial solution x = 0, y = 0, z = 0c)it can be reduced to a single equation and so a solution does not existd)determinant of the matrix of coefficients is zeroCorrect answer is option 'B'. Can you explain this answer?

Question

Answer

Understanding the System of Equations
The given system of equations is:
1. 3x + 2y + z = 0
2. x + 4y + z = 0
3. 2x + y + 4z = 0
To analyze this system, we can represent it in matrix form and determine its solution.
Matrix Representation
The equations can be represented in matrix form as:
$$
\begin{bmatrix}
3 & 2 & 1 \
1 & 4 & 1 \
2 & 1 & 4
\end{bmatrix}
\begin{bmatrix}
x \
y \
z
\end{bmatrix}
=
\begin{bmatrix}
0 \
0 \
0
\end{bmatrix}
$$
Calculating the Determinant
To find if the system has only the trivial solution, we calculate the determinant of the coefficient matrix:
$$
\text{det} = 3(4 \cdot 4 - 1 \cdot 1) - 2(1 \cdot 4 - 1 \cdot 2) + 1(1 \cdot 1 - 4 \cdot 2)
$$
Calculating this, we find that the determinant is non-zero, indicating that the system has only the trivial solution.
Conclusion
Since the determinant is non-zero, the only solution to the equations is:
- x = 0
- y = 0
- z = 0
Thus, the correct answer is option 'B': it has only the trivial solution x = 0, y = 0, z = 0.

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