If $\mathrm{P}=\left[\begin{array}{cc}1 & 0 \\ \frac{1}{2} & 1\end{array}\right]$, then $\mathrm{P}^{50}$ is :

(A) $\left[\begin{array}{cc}1 & 0 \\ 25 & 1\end{array}\right]$

(B) $\left[\begin{array}{cc}1 & 25 \\ 0 & 1\end{array}\right]$

(C) $\left[\begin{array}{cc}1 & 0 \\ 50 & 1\end{array}\right]$

(D) $\left[\begin{array}{cc}1 & 50 \\ 0 & 1\end{array}\right]$

JEE Main · 2021 · 25 Jul 2021 · Shift 2 · Matrices

P = [1 \frac{1}{2} 01]

, then

P^{50}

is :

Answer: (A)

Computing powers of

P

P^{2} = [1 \frac{1}{2} 01] [1 \frac{1}{2} 01] = [1101]

P^{3} = P^{2} \cdot P = [1 \frac{3}{2} 01]

By induction, we can show that

P^{n} = [1 \frac{n}{2} 01]

.

Therefore,

P^{50} = [12501]

JEE Main · 2021 · 25 Jul 2021 · Shift 2 · Single correct option