The system of linear equations `x+lambday-z=0` `lambdax-y-z=0` `x+y-lambdaz=0` has a non-trivial solution for : (1) infinitely many values of `lambda` . (2) exactly one value of `lambda` . (3) exactly two values of `lambda` . (4) exactly three values of `lambda` .

Explanation

To determine the values of \( \lambda \) for which the given system of linear equations has a non-trivial solution, we start by writing the system of equations in matrix form and finding the determinant of the coefficient matrix. The equations given are:

1. \( x + \lambda y - z = 0 \)
2. \( \lambda x - y - z = 0 \)
3. \( x + y - \lambda z = 0 \)

### Step 1: Write the coefficient matrix
The coefficient matrix \( A \) for the system can be written as:
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