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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 $M=\left(\begin{array}{cc}\frac{5}{2} & \frac{3}{2} \\ -\frac{3}{2} & -\frac{1}{2}\end{array}\right)$, then which of the following matrices is equal to $M^{2022}$ ?
  • ✓
    $\left(\begin{array}{cc}3034 & 3033 \\ -3033 & -3032\end{array}\right)$
  • B
     $\left(\begin{array}{ll}3034 & -3033 \\ 3033 & -3032\end{array}\right)$
  • C
    $\left(\begin{array}{cc}3033 & 3032 \\ -3032 & -3031\end{array}\right)$
  • D
    $\left(\begin{array}{cc}3032 & 3031 \\ -3031 & -3030\end{array}\right)$
✓

Answer

Correct option: A.
$\left(\begin{array}{cc}3034 & 3033 \\ -3033 & -3032\end{array}\right)$
a
$\begin{array}{l}M=\left[\begin{array}{cc}\frac{5}{2} & \frac{3}{2} \\ \frac{-3}{2} & \frac{-1}{2}\end{array}\right] \\ M=\left[\begin{array}{cc}\frac{3}{2}+1 & \frac{3}{2} \\ \frac{-3}{2} & \frac{-3}{2}+1\end{array}\right] \\ M = I +\frac{3}{2}\left[\begin{array}{cc}1 & 1 \\ -1 & -1\end{array}\right] \\ \text { Let } A=\left[\begin{array}{cc}1 & 1 \\ -1 & -1\end{array}\right] \\ A^2=\left[\begin{array}{cc}1 & 1 \\ -1 & -1\end{array}\right]\left[\begin{array}{cc}1 & 1 \\ -1 & -1\end{array}\right]=\left[\begin{array}{ll}0 & 0 \\ 0 & 0\end{array}\right] \\ M ^{2022}=\left( I +\frac{3}{2} A \right)^{20022} \\ = I +3033 A \\ =\left[\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right]+3033\left[\begin{array}{cc}1 & 1 \\ -1 & -1\end{array}\right] \\ =\left[\begin{array}{cc}3034 & 3033 \\ -3033 & -3032\end{array}\right] \\\end{array}$

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