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3 and 4 . determinant and metrices — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMaths3 and 4 . determinant and metrices1 Mark
MCQ
If $P=\left[\begin{array}{ll}1 & 0 \\ 1 / 2 & 1\end{array}\right]$, then $P^{50}$ is:
  • A
    $\left[\begin{array}{cc}1 & 25 \\ 0 & 1\end{array}\right]$
  • ✓
    $\left[\begin{array}{ll}1 & 0 \\ 25 & 1\end{array}\right]$
  • C
    $\left[\begin{array}{ll}1 & 0 \\ 50 & 1\end{array}\right]$
  • D
    $\left[\begin{array}{cc}1 & 50 \\ 0 & 1\end{array}\right]$
✓

Answer

Correct option: B.
$\left[\begin{array}{ll}1 & 0 \\ 25 & 1\end{array}\right]$
b
$P=\left[\begin{array}{cc}1 & 0 \\ \frac{1}{2} & 1\end{array}\right]$

$P^{2}=\left[\begin{array}{ll}1 & 0 \\ \frac{1}{2} & 1\end{array}\right]\left[\begin{array}{ll}1 & 0 \\ \frac{1}{2} & 1\end{array}\right]=\left[\begin{array}{ll}1 & 0 \\ 1 & 1\end{array}\right]$

$P^{3}=\left[\begin{array}{ll}1 & 0 \\ 1 & 1\end{array}\right]\left[\begin{array}{cc}1 & 0 \\ \frac{1}{2} & 1\end{array}\right]=\left[\begin{array}{cc}1 & 0 \\ \frac{3}{2} & 1\end{array}\right]$

$P^{4}=\left[\begin{array}{ll}1 & 0 \\ 1 & 1\end{array}\right]\left[\begin{array}{ll}1 & 0 \\ 1 & 1\end{array}\right]=\left[\begin{array}{ll}1 & 0 \\ 2 & 1\end{array}\right]$

$\vdots$

$\therefore P^{50}=\left[\begin{array}{cc}1 & 0 \\ 25 & 1\end{array}\right]$

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