IIT-JEE 2011 · Paper 2 · Mathematics Q43

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MathematicsSingle correct+3 / -1Question 43 of 60

Let $\omega\ne1$ 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

Answer and worked solution

Official answer: A

Determinant. Expanding and using $\omega^3=1$ :
$\Delta=1-c\omega-a\omega+ac\omega^2=(1-a\omega)(1-c\omega).$

Non-singular needs $a\omega\ne1$ and $c\omega\ne1$ , i.e. $a\ne\omega^2$ and $c\ne\omega^2$ . So $a=c=\omega$ , while $b$ is free (2 choices): 2 matrices.

(A)

Topics:Determinants and properties Roots of unity

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IIT-JEE 2011 · Paper 2, Mathematics question 43