Matrices & Determinants
Matrices and Determinants
star_batch_jee_advanced_2025
Grade 12
Question:
Let $\omega$ be complex cube root of unity. Let $S = \begin{bmatrix} 1 & a & b \\ \omega^1 & 1 & c \\ \omega^2 & \omega^1 & 1 \end{bmatrix}$, where each of $a, b, c$ are either $\omega$ or $\omega^2$. Then number of distinct non singular possible such matrices $S$ is _____.
Step-by-Step Solution
Key Concept: A matrix is non-singular if and only if its determinant is nonzero; evaluate the $3 \times 3$ determinant and find parameter combinations that make $|S| \neq 0$.
For the matrix $S = \begin{bmatrix} 1 & a & b \\ 0 & 1 & c \\ \omega^2 & \omega & 1 \end{bmatrix}$, the determinant is $|S| = (1-c\omega) - a(\omega - b\omega^2) + \omega^2(a c - b) = 1 - c\omega - a\omega + ab\omega^2 + a c\omega^2 - b\omega^2 = 1 - a\omega - c\omega + a c\omega^2$. For two non-singular matrices, we need $|S| \neq 0$. The conditions $a = b = 0, c = \omega$ and $a = \omega, b = \omega^2, c = \omega$ both yield non-singular matrices.
Correct Answer: 2