If M = ( 5 2 3 2 - 3 2 - 1 2 ) , then which of the following matrices is equal to M 2022 ? [2022]

Question

If $M = (\begin{matrix} \frac{5}{2} & \frac{3}{2} \\ - \frac{3}{2} & - \frac{1}{2} \end{matrix}),$ then which of the following matrices is equal to $M^{2022} ?$ [2022]

Smart Achievers · Accepted Answer

(1)

$Given that M = (\begin{matrix} \frac{5}{2} & \frac{3}{2} \\ - \frac{3}{2} & - \frac{1}{2} \end{matrix}) \Rightarrow M^{2} = (\begin{matrix} 4 & 3 \\ - 3 & - 2 \end{matrix})$

$\Rightarrow M^{3} = (\begin{matrix} \frac{11}{2} & \frac{9}{2} \\ - \frac{9}{2} & - \frac{7}{2} \end{matrix}) \Rightarrow M^{4} = (\begin{matrix} 7 & 6 \\ - 6 & - 5 \end{matrix}) and so on$

$\Rightarrow M^{n} = (\begin{matrix} \frac{3 n}{2} + 1 & \frac{3 n}{2} \\ - \frac{3 n}{2} & - \frac{3 n}{2} + 1 \end{matrix})$

$Now, M^{n} = (\begin{matrix} \frac{3 \times 2022}{2} + 1 & \frac{3 \times 2022}{2} \\ - \frac{3 \times 2022}{2} & - \frac{3 \times 2022}{2} + 1 \end{matrix})$

$= (\begin{matrix} 3034 & 3033 \\ - 3033 & - 3032 \end{matrix})$

$∴$ $Option (1) is correct$

Solution

Q.
If $M = (\begin{matrix} \frac{5}{2} & \frac{3}{2} \\ - \frac{3}{2} & - \frac{1}{2} \end{matrix}),$ then which of the following matrices is equal to $M^{2022}$ [2022]

1 $(\begin{matrix} 3034 & 3033 \\ - 3033 & - 3032 \end{matrix})$

2 $(\begin{matrix} 3034 & - 3033 \\ 3033 & - 3032 \end{matrix})$

3 $(\begin{matrix} 3033 & - 3032 \\ - 3032 & - 3031 \end{matrix})$

4 $(\begin{matrix} 3032 & 3031 \\ - 3031 & - 3030 \end{matrix})$

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Solution

Q. If M=(5232-32-12), then which of the following matrices is equal to M2022 [2022]

1 (30343033-3033-3032)

2 (3034-30333033-3032)

3 (3033-3032-3032-3031)

4 (30323031-3031-3030)