If the equations `ax+by + cz=0, bx + cy + az=0 and cx + ay + bz=0` have a non-zero 'solution, then which one of the following is true

Options

A. `a+b+c=0`
B. ` a=b=0`
C. `(a-b)+(b-c)^2 +(c - a)^2 =0`
D. none of these

Answer

To determine the condition under which the equations \( ax + by + cz = 0 \), \( bx + cy + az = 0 \), and \( cx + ay + bz = 0 \) have a non-zero solution, we can analyze the system of equations using determinants.

### Step-by-Step Solution:

1. **Write the equations in matrix form:**
We can express the given equations in matrix form as follows:
\[ \begin{bmatrix} a & b & c \\ b & c & a \\ c & a & b \end{bmatrix} \begin{bmatrix} x \\ y \\ z \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \\ 0 \end{bmatrix} \]

2. **Determine the condition for non-zero solutions:**
For this system of equations to have non-zero solutions, the determinant of the coefficient matrix must be zero:
\[ \text{det} \begin{bmatrix} a & b & c \\ b & c & a \\ c & a & b \end{bmatrix} = 0 \]

3. **Calculate the determinant:**
The determinant can be calculated using the formula for a 3x3 matrix:
\[ \text{det}(A) = a(ei - fh) - b(di - fg) + c(dh - eg) \]
Applying this to our matrix:
\[ \text{det} = a(c \cdot b - a \cdot a) - b(b \cdot b - c \cdot a) + c(b \cdot a - c \cdot c) \]
Simplifying this gives:
\[ = abc - a^3 - b^3 + abc + cba - c^3 = 3abc - (a^3 + b^3 + c^3) \]

4. **Set the determinant to zero:**
For a non-zero solution, we set the determinant to zero:
\[ 3abc - (a^3 + b^3 + c^3) = 0 \]

5. **Factor the equation:**
Rearranging gives:
\[ a^3 + b^3 + c^3 - 3abc = 0 \]
This can be factored using the identity:
\[ a^3 + b^3 + c^3 - 3abc = (a + b + c)(a^2 + b^2 + c^2 - ab - ac - bc) \]

6. **Analyze the factors:**
For the product to be zero, at least one of the factors must be zero:
- \( a + b + c = 0 \)
- \( a^2 + b^2 + c^2 - ab - ac - bc = 0 \) (which implies \( a = b = c \))

Thus, the condition for the equations to have a non-zero solution is that either \( a + b + c = 0 \) or \( a = b = c \).

### Conclusion:
The correct answer is that for the equations to have a non-zero solution, it is true that:
- \( a + b + c = 0 \) or \( a = b = c \).