Animated Solution for Mathematics - Matrices and Determinants: Let $\omega$ be a complex cube root of unity with $\omega \ne 1$ and $P=[p_{ij}]$ be an $n\times n$ matrix with $p_{ij}={\omega }^{i+j}$. Then $P^2\ne 0$, when $n=$

We begin with a matrix $P$ where each element is defined by $p_{ij}={\omega }^{i+j}$. Here, $\omega$ is a complex cube root of unity, which provides the fundamental property ${\omega }^3=1$.

To understand the behavior of $P^2$, we perform matrix multiplication. The $(i,j)$ entry of $P^2$ is defined by the summation:

By applying the laws of exponents, the expression simplifies to:

We can factor out the terms independent of the summation index $k$:

Let $S={\sum }_{k=1}^n({\omega }^2{)}^k$. This is a geometric progression where the first term $a={\omega }^2$ and the common ratio $r={\omega }^2$.

Using the standard sum formula for a geometric series, $S=a\frac{r^n-1}{r-1}$, we obtain:

For the matrix $P^2$ to be the zero matrix, the sum $S$ must equal zero. This occurs when $({\omega }^2{)}^n=1$, or equivalently, ${\omega }^{2n}=1$.

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Given the property ${\omega }^3=1$, the condition ${\omega }^{2n}=1$ is satisfied if and only if $2n$ is a multiple of $3$. Since $2$ and $3$ are coprime, this implies that $n$ must be a multiple of $3$.

The problem asks for the condition where $P^2 eq 0$. This occurs if and only if $n$ is not a multiple of $3$.

If we are evaluating specific values for $n$, we identify that any value not divisible by $3$ satisfies the condition. For instance, if $n=57$, $P^2=0$, whereas for $n\in \{55,56,58\}$, the condition $P^2 eq 0$ holds true.