Matrices & Determinants
Matrices and Determinants
star_batch_jee_advanced_2025
Grade 12
Question:
Let $\omega$ be a complex cube root of unity with $\omega \neq 1$ and $P = \left[p_{ij}\right]$ be a $n\times n$ matrix with $p_{ij} = \omega^{i+j}$. Then $P^2 \neq 0$ when $n =$
Step-by-Step Solution
Key Concept: The matrix structure depends critically on whether $n$ is a multiple of 3, which determines linear dependence of rows.
For $n=1$: $p = [w^2]$ gives $p^2 = [w^4] \neq 0$. For $n=2$: the determinant involves $w^4 + 1$ terms which is nonzero. For $n=3$: the matrix becomes circulant with the last row containing powers of $w$, and the determinant equals zero when rows become linearly dependent. Similarly, $p^2 \neq 0$ when $n$ is not a multiple of 3.
Correct Answer: 2,3,4