If `|(p,-q,0),(0,p,q),(q,0,p)|=0`, then which one of the following is correct ?

Options

A. A. p is one of the cube roots of unity
B. B. q is one of the cube roots of unity
C. C. `(p)/(q)` is one of the cube roots of unity
D. D. None of the above

Answer

To solve the problem, we need to find the determinant of the matrix given and analyze the condition that it equals zero. The matrix is:

\[ \begin{vmatrix} p & -q & 0 \\ 0 & p & q \\ q & 0 & p \end{vmatrix} \]

### Step 1: Calculate the determinant

We will calculate the determinant using the method of cofactor expansion along the first row.

\[ \text{Determinant} = p \cdot \begin{vmatrix} p & q \\ 0 & p \end{vmatrix} - (-q) \cdot \begin{vmatrix} 0 & q \\ q & p \end{vmatrix} + 0 \cdot \begin{vmatrix} 0 & p \\ q & 0 \end{vmatrix} \]

### Step 2: Calculate the 2x2 determinants

1. Calculate the first 2x2 determinant:
\[ \begin{vmatrix} p & q \\ 0 & p \end{vmatrix} = p \cdot p - q \cdot 0 = p^2 \]

2. Calculate the second 2x2 determinant:
\[ \begin{vmatrix} 0 & q \\ q & p \end{vmatrix} = 0 \cdot p - q \cdot q = -q^2 \]

### Step 3: Substitute back into the determinant expression

Now substituting back into the determinant expression:

\[ \text{Determinant} = p \cdot p^2 + q \cdot (-q^2) = p^3 - q^3 \]

### Step 4: Set the determinant equal to zero

Given that the determinant is equal to zero, we have:

\[ p^3 - q^3 = 0 \]

### Step 5: Factor the equation

This can be factored as:

\[ (p - q)(p^2 + pq + q^2) = 0 \]

### Step 6: Analyze the factors

From this equation, we have two cases:

1. \( p - q = 0 \) which implies \( p = q \).
2. \( p^2 + pq + q^2 = 0 \) which does not yield real solutions unless \( p \) and \( q \) are complex numbers.

### Step 7: Conclude the relation between p and q

Since we are looking for cube roots of unity, we can conclude that \( p^3 = q^3 \) implies:

\[ \frac{p^3}{q^3} = 1 \implies \left(\frac{p}{q}\right)^3 = 1 \]

This means \( \frac{p}{q} \) is one of the cube roots of unity.

### Final Answer

The correct option is:
C. \( \frac{p}{q} \) is one of the cube roots of unity.

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