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Q.

Let A be a $2 \times 2$ matrix with rational entries such that $\det (A^{2} - 2 I_{2}) = 0$ where det or $| . |$ represents determinant and $I_{2}$ is $2 \times 2$ identity matrix. Choose correct option(s).

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a

$A^{99} = 2^{49} A$

b

$| A^{9} | = 2^{9}$

c

$A^{100} = 2^{50} A$

d

$| A^{40} | = 2^{40}$

answer is B, C.

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Detailed Solution

$\det (A^{2} - 2 I) = 0 \Rightarrow \det (A - \sqrt{2} I) \det (A + \sqrt{2} I) = 0$
As A contains rational entries, $\det (A - \sqrt{2} I) = \det (A + \sqrt{2} I) = 0$
So characteristics equation of A is $A^{2} - 2 I_{2} = 0$
$A^{2 n} = 2^{n} I_{2}, A^{2 n + 1} = 2^{n} A & | A | = - 2.$

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