AP EAMCET · Maths · Matrices
Let \(1, \omega\) and \(\omega^2\) be the cube roots of unity. If \(\mathrm{S}\) is the set of all non-singular matrices of the form \(\left[\begin{array}{rrr}1 & a & b \\ \omega & 1 & c \\ \omega^2 & \omega & 1\end{array}\right]\) where \(a, b, c \in\left\{\omega, \omega^2\right\}\), then the number of elements in \(\mathrm{S}\) is
- A 2
- B 3
- C 4
- D 6
Answer & Solution
Correct Answer
(A) 2
Step-by-step Solution
Detailed explanation
\(\det(M) = \begin{vmatrix} 1 & a & b \\ \omega & 1 & c \\ \omega^2 & \omega & 1 \end{vmatrix}\) \(\det(M) = 1(1-c\omega) - a(\omega - c\omega^2) + b(\omega^2 - \omega^2)\) \(\det(M) = 1 - c\omega - a\omega + ac\omega^2\) Given \(a, b, c \in \{\omega, \omega^2\}\). Case 1:…
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