If `x + y +z=0, 4x+3y -z=0 and 3x + 5y +3z=0` is the given system of equations, then which of the following is correct?

Options

A. it is inconsistant
B. it has only single solution x = 0, y =0, z =0
C. determinant of coefficients of matrix is zero
D. none of these

Answer

To solve the given system of equations:

1. **Write down the equations:**
\[ \begin{align*} (1) & \quad x + y + z = 0 \\ (2) & \quad 4x + 3y - z = 0 \\ (3) & \quad 3x + 5y + 3z = 0 \end{align*} \]

2. **Form the coefficient matrix:**
The coefficient matrix \( A \) for the system of equations is:
\[ A = \begin{bmatrix} 1 & 1 & 1 \\ 4 & 3 & -1 \\ 3 & 5 & 3 \end{bmatrix} \]

3. **Calculate the determinant of the coefficient matrix:**
We will calculate the determinant \( |A| \) using the formula for a 3x3 matrix:
\[ |A| = a(ei - fh) - b(di - fg) + c(dh - eg) \]
where the matrix is:
\[ \begin{bmatrix} a & b & c \\ d & e & f \\ g & h & i \end{bmatrix} \]
For our matrix:
\[ |A| = 1 \cdot (3 \cdot 3 - (-1) \cdot 5) - 1 \cdot (4 \cdot 3 - (-1) \cdot 3) + 1 \cdot (4 \cdot 5 - 3 \cdot 3) \]

Now calculating each term:
\[ |A| = 1 \cdot (9 + 5) - 1 \cdot (12 + 3) + 1 \cdot (20 - 9) \]
\[ |A| = 1 \cdot 14 - 1 \cdot 15 + 1 \cdot 11 \]
\[ |A| = 14 - 15 + 11 = 10 \]

4. **Determine the nature of the solution:**
Since the determinant \( |A| = 10 \) is not equal to 0, the system of equations has a unique solution.

5. **Conclusion:**
Therefore, the correct option is that the system has only one single solution, which is \( x = 0, y = 0, z = 0 \).