Let `omega` be a complex cube root of unity with `omega ne 0` and `P=[p_(ij)]` be an n x n matrix with `p_(ij)=omega^(i+j)`. Then `p^2ne0` when n is equal to :

Options

A. 57
B. 55
C. 58
D. 56

Answer

To solve the problem, we need to analyze the given matrix \( P = [p_{ij}] \) where \( p_{ij} = \omega^{i+j} \) and \( \omega \) is a complex cube root of unity. We want to find when \( P^2 \neq 0 \).

### Step-by-step Solution:

1. **Understanding the Matrix \( P \)**:
The matrix \( P \) is defined such that each element is given by \( p_{ij} = \omega^{i+j} \). This means that the element in the \( i \)-th row and \( j \)-th column is the complex number \( \omega \) raised to the power of \( i+j \).

2. **Structure of the Matrix**:
For a \( n \times n \) matrix, the first few elements will look like:
\[ P = \begin{bmatrix} \omega^{2} & \omega^{3} & \omega^{4} & \ldots & \omega^{n+1} \\ \omega^{3} & \omega^{4} & \omega^{5} & \ldots & \omega^{n+2} \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ \omega^{n+1} & \omega^{n+2} & \omega^{n+3} & \ldots & \omega^{2n} \end{bmatrix} \]

3. **Calculating \( P^2 \)**:
To find \( P^2 \), we need to compute the product \( P \times P \). The element in the \( (i,k) \)-th position of \( P^2 \) is given by:
\[ (P^2)_{ik} = \sum_{j=1}^{n} p_{ij} \cdot p_{jk} = \sum_{j=1}^{n} \omega^{i+j} \cdot \omega^{j+k} = \sum_{j=1}^{n} \omega^{i + 2j + k} \]
This simplifies to:
\[ (P^2)_{ik} = \omega^{i+k} \sum_{j=1}^{n} \omega^{2j} = \omega^{i+k} \cdot \frac{\omega^2(1 - \omega^{2n})}{1 - \omega^2} \]
where the sum \( \sum_{j=1}^{n} \omega^{2j} \) is a geometric series.

4. **Condition for \( P^2 \neq 0 \)**:
The matrix \( P^2 \) will be the zero matrix if the sum \( \sum_{j=1}^{n} \omega^{2j} = 0 \). This occurs when \( n \) is a multiple of 3, because \( \omega^2 \) is a cube root of unity and the sum of the first \( n \) terms will be zero if \( n \) is a multiple of 3.

5. **Conclusion**:
Therefore, \( P^2 \neq 0 \) when \( n \) is not a multiple of 3. The values of \( n \) for which \( P^2 \neq 0 \) are all integers except those that are multiples of 3.

### Final Answer:
\( P^2 \neq 0 \) when \( n \) is not a multiple of 3.