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JEE Main 202125 JulyEvening Shift
Exam
JEE Main
Year
2021
Date
25 July
Shift
Evening Shift
Subject
Mathematics
Chapter
Matrices

JEE Main 2021 · 25 July · Evening Shift · Mathematics · Matrices

If P = [ cases 1 & 0 (1)/(2) & 1 cases ], then P ^ 50 is:

If P=P =\left[{10121\begin{cases}1 & 0 \\ \dfrac{1}{2} & 1\end{cases}\right],thenthen{P}^{50} is:

Options

  1. A\left[\begin{cases}1 & 0 \\ 25 & 1\end{cases}\right]
  2. B\left[\begin{cases}1 & 50 \\ 0 & 1\end{cases}\right]
  3. C\left[\begin{cases}1 & 25 \\ 0 & 1\end{cases}\right]
  4. D\left[\begin{cases}1 & 0 \\ 50 & 1\end{cases}\right]

Correct answer

A. \left[\begin{cases}1 & 0 \\ 25 & 1\end{cases}\right]

Step-by-step solution

P = \left[\begin{cases}1 & 0 \\ \dfrac{1}{2} & 1\end{cases}\right]

{P}^{2} = \left[\begin{cases}1 & 0 \\ \dfrac{1}{2} & 1\end{cases}\right]\left[\begin{cases}1 & 0 \\ \dfrac{1}{2} & 1\end{cases}\right] = \left[\begin{cases}1 & 0 \\ 1 & 1\end{cases}\right]

{P}^{3} = \left[\begin{cases}1 & 0 \\ 1 & 1\end{cases}\right]\left[\begin{cases}1 & 0 \\ \dfrac{1}{2} & 1\end{cases}\right] = \left[\begin{cases}1 & 0 \\ \dfrac{3}{2} & 1\end{cases}\right]

{P}^{4} = \left[\begin{cases}1 & 0 \\ \dfrac{1}{2} & 1\end{cases}\right]\left[\begin{cases}1 & 0 \\ \dfrac{3}{2} & 1\end{cases}\right] = \left[\begin{cases}1 & 0 \\ 2 & 1\end{cases}\right]

⋮⋮

∴ {P}^{50} = \left[\begin{cases}1 & 0 \\ \dfrac{1}{2} + 49\left(\dfrac{1}{2}\right) & 1\end{cases}\right] = \left[\begin{cases}1 & 0 \\ 25 & 1\end{cases}\right]

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