JEE Mains · Maths · STD 12 - 3 and 4 . metrices and determinant
Let \(A =\left[\begin{array}{ccc}2 & -1 & -1 \\ 1 & 0 & -1 \\ 1 & -1 & 0\end{array}\right]\) and \(B = A - I\). If \(\omega=\frac{\sqrt{3} i -1}{2}\) then the number of elements in the set \(\left\{ n \in\{1,2, \ldots, 100\}: A ^{ n }+(\omega B )^{ n }= A + B \right\}\) is equal to \(..........\)
- A \(17\)
- B \(15\)
- C \(14\)
- D \(13\)
Answer & Solution
Correct Answer
(A) \(17\)
Step-by-step Solution
Detailed explanation
\(A=\left[\begin{array}{ccc}2 & -1 & -1 \\ 1 & 0 & -1 \\ 1 & -1 & 0\end{array}\right] \Rightarrow A^{2}=A \Rightarrow A^{n}=A\) Now, \(B = A - I =\left[\begin{array}{lll}1 & -1 & -1 \\ 1 & -1 & -1 \\ 1 & -1 & -1\end{array}\right]\) \(B ^{2}=- B\) \(B ^{3}=- B ^{2}= B\)…
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