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Multiplication of Matrices

Let P be a sq...

Question

Let $P$ be a square matrix satisfying $P^{2} = I - P,$ where $I$ is identity matrix. If $P^{n} = 5 I - 8 P,$ then the value of $n$ is

5

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6

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7

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8

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Solution

The correct option is B $6$
$P^{2} = I - P$
$\Rightarrow P^{3} = P I - P^{2} \Rightarrow P^{3} = P - (I - P) = - I + 2 P \Rightarrow P^{4} = 2 P^{2} - P \Rightarrow P^{4} = 2 (I - P) - P = 2 I - 3 P$

$P^{4} = P^{2} - P^{3} \Rightarrow P^{5} = P^{3} - P^{4} \Rightarrow P^{5} = - 3 I + 5 P P^{6} = P^{4} - P^{5} \Rightarrow P^{6} = 5 I - 8 P ∴ n = 6$

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P^{2} = I - P,

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