Given that the system of equations `x=cy+bz ,y=az+cx , z=bx +ay` has nonzero solutions and and at least one of the a,b,c is a proper fraction.
`a^(2)+b^(2)+c^(2)` is

Options

A. ` gt 2`
B. `gt 3`
C. `lt 3`
D. `lt2`

Explanation

To solve the given system of equations and find the value of \( a^2 + b^2 + c^2 \), we will follow these steps:

### Step 1: Write the system of equations in standard form
The given equations are:
1. \( x = cy + bz \)
2. \( y = az + cx \)
3. \( z = bx + ay \)

Rearranging them gives:
1. \( x - cy - bz = 0 \)
2. \( -cx + y - az = 0 \)
3. \( -bx - ay + z = 0 \)

### Step 2: Set up the coefficient matrix
From the rearranged equations, we can form the coefficient matrix \( A \):
\[ A = \begin{pmatrix} 1 & -c & -b \\ -c & 1 & -a \\ -b & -a & 1 \end{pmatrix} \]

### Step 3: Find the determinant of the matrix
For the system to have non-zero solutions, the determinant of the matrix must be zero:
\[ \text{det}(A) = 0 \]

Calculating the determinant:
\[ \text{det}(A) = 1 \cdot \begin{vmatrix} 1 & -a \\ -a & 1 \end{vmatrix} - (-c) \cdot \begin{vmatrix} -c & -a \\ -b & 1 \end{vmatrix} - (-b) \cdot \begin{vmatrix} -c & 1 \\ -b & -a \end{vmatrix} \]

Calculating the 2x2 determinants:
1. \( \begin{vmatrix} 1 & -a \\ -a & 1 \end{vmatrix} = 1 - a^2 \)
2. \( \begin{vmatrix} -c & -a \\ -b & 1 \end{vmatrix} = -c + ab \)
3. \( \begin{vmatrix} -c & 1 \\ -b & -a \end{vmatrix} = ac - b \)

Substituting back:
\[ \text{det}(A) = 1(1 - a^2) + c(c - ab) + b(ac - b) \]
Expanding this gives:
\[ \text{det}(A) = 1 - a^2 + c^2 - abc + acb - b^2 = 1 - a^2 - b^2 - c^2 + 2abc \]

Setting the determinant to zero:
\[ 1 - a^2 - b^2 - c^2 + 2abc = 0 \]
Thus, we have:
\[ a^2 + b^2 + c^2 + 2abc = 1 \]

### Step 4: Analyze the conditions
Given that at least one of \( a, b, c \) is a proper fraction, we know:
- \( 1 - a^2 > 0 \)
- \( 1 - b^2 > 0 \)
- \( 1 - c^2 > 0 \)

This implies:
\[ a^2 < 1, \quad b^2 < 1, \quad c^2 < 1 \]

### Step 5: Sum the inequalities
Adding these inequalities:
\[ (1 - a^2) + (1 - b^2) + (1 - c^2) > 0 \implies 3 - (a^2 + b^2 + c^2) > 0 \]
Thus:
\[ a^2 + b^2 + c^2 < 3 \]

### Conclusion
The value of \( a^2 + b^2 + c^2 \) is less than 3.