JEE Advanced 2011 Paper 2Question 1
Let $\omega\neq 1$ be a cube root of unity and $S$ be the set of all non-singular matrices of the form $$\begin{bmatrix}1&a&b\\ \omega&1&c\\ \omega^2&\omega&1\end{bmatrix},$$ where each of $a$, $b$, and $c$ is either $\omega$ or $\omega^2$. Then the number of distinct matrices in the set $S$ is
Determinant
Solution
$\det=1-\omega(a+c)+ac\,\omega^2$. For $a=c=\omega$: $-3\omega^2\ne0$. For $a=c=\omega^2$: $1-2+1=0$. For $a\ne c$: $a+c=-1$, $ac=1$ so $1+\omega+\omega^2=0$. So only $a=c=\omega$ with $b\in\{\omega,\omega^2\}$ works: $2$ matrices.